Geometry & Topology Articles (Project Euclid)
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The latest articles from Geometry & Topology on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 14:27 EDTThu, 19 Oct 2017 14:27 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On representation varieties of $3$–manifold groups
https://projecteuclid.org/euclid.gt/1508437634
<strong>Michael Kapovich</strong>, <strong>John Millson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 4, 1931--1968.</p><p><strong>Abstract:</strong><br/>
We prove universality theorems (“Murphy’s laws”) for representation varieties of fundamental groups of closed [math] –dimensional manifolds. We show that germs of [math] –representation schemes of such groups are essentially the same as germs of schemes over [math] of finite type.
</p>projecteuclid.org/euclid.gt/1508437634_20171019142734Thu, 19 Oct 2017 14:27 EDTSharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature boundshttps://projecteuclid.org/euclid.gt/1510859141<strong>Fabio Cavalletti</strong>, <strong>Andrea Mondino</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 1, 603--645.</p><p><strong>Abstract:</strong><br/> For metric measure spaces satisfying the reduced curvature–dimension condition [math] we prove a series of sharp functional inequalities under the additional “essentially nonbranching” assumption. Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and, more generally, [math] spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds. In particular we prove the Brunn–Minkowski inequality, the [math] –spectral gap (or equivalently the [math] –Poincaré inequality) for any [math] , the log-Sobolev inequality, the Talagrand inequality and finally the Sobolev inequality. All the results are proved in a sharp form involving an upper bound on the diameter of the space; all our inequalities for essentially nonbranching [math] spaces take the same form as the corresponding sharp ones known for a weighted Riemannian manifold satisfying the curvature–dimension condition [math] in the sense of Bakry and Émery. In this sense our inequalities are sharp. We also discuss the rigidity and almost rigidity statements associated to the [math] –spectral gap. In particular, we have also shown that the sharp Brunn–Minkowski inequality in the global form can be deduced from the local curvature–dimension condition, providing a step towards (the long-standing problem of) globalization for the curvature–dimension condition [math] . To conclude, some of the results can be seen as answers to open problems proposed in Villani’s book Optimal transport . </p>projecteuclid.org/euclid.gt/1510859141_20171116140542Thu, 16 Nov 2017 14:05 ESTLimits of limit sets, II: Geometrically infinite groupshttps://projecteuclid.org/euclid.gt/1510859167<strong>Mahan Mj</strong>, <strong>Caroline Series</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 647--692.</p><p><strong>Abstract:</strong><br/>
We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon–Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences, we show that there exist examples where even pointwise convergence of Cannon–Thurston maps fails.
</p>projecteuclid.org/euclid.gt/1510859167_20171116140618Thu, 16 Nov 2017 14:06 ESTMaximally stretched laminations on geometrically finite hyperbolic manifoldshttps://projecteuclid.org/euclid.gt/1510859168<strong>François Guéritaud</strong>, <strong>Fanny Kassel</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 693--840.</p><p><strong>Abstract:</strong><br/>
Let [math] be a discrete group. For a pair [math] of representations of [math] into [math] with [math] geometrically finite, we study the set of [math] –equivariant Lipschitz maps from the real hyperbolic space [math] to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is “maximally stretched” by all such maps when the minimal constant is at least [math] . As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups [math] of [math] on [math] by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups [math] the action remains properly discontinuous after any small deformation of [math] inside [math] .
</p>projecteuclid.org/euclid.gt/1510859168_20171116140618Thu, 16 Nov 2017 14:06 ESTAnalytic nonabelian Hodge theoryhttps://projecteuclid.org/euclid.gt/1510859169<strong>Jonathan Pridham</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 841--902.</p><p><strong>Abstract:</strong><br/>
The proalgebraic fundamental group can be understood as a completion with respect to finite-dimensional noncommutative algebras. We introduce finer invariants by looking at completions with respect to Banach and [math] –algebras, from which we can recover analytic and topological representation spaces, respectively. For a compact Kähler manifold, the [math] –completion also gives the natural setting for nonabelian Hodge theory; it has a pure Hodge structure, in the form of a pro- [math] –dynamical system. Its representations are pluriharmonic local systems in Hilbert spaces, and we study their cohomology, giving a principle of two types, and splittings of the Hodge and twistor structures.
</p>projecteuclid.org/euclid.gt/1510859169_20171116140618Thu, 16 Nov 2017 14:06 ESTModular operads of embedded curveshttps://projecteuclid.org/euclid.gt/1510859170<strong>Satoshi Kondo</strong>, <strong>Charles Siegel</strong>, <strong>Jesse Wolfson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 903--922.</p><p><strong>Abstract:</strong><br/>
For each [math] , we construct a modular operad [math] of “ [math] –log-canonically embedded” curves. We also construct, for [math] , a stable cyclic operad [math] of such curves, and, for [math] , a cyclic operad [math] of “ [math] –log-canonically embedded” rational curves.
</p>projecteuclid.org/euclid.gt/1510859170_20171116140618Thu, 16 Nov 2017 14:06 ESTEquidistribution for sequences of line bundles on normal Kähler spaceshttps://projecteuclid.org/euclid.gt/1510859171<strong>Dan Coman</strong>, <strong>Xiaonan Ma</strong>, <strong>George Marinescu</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 923--962.</p><p><strong>Abstract:</strong><br/>
We study the asymptotics of Fubini–Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kähler complex space.
</p>projecteuclid.org/euclid.gt/1510859171_20171116140618Thu, 16 Nov 2017 14:06 ESTExistence of Lefschetz fibrations on Stein and Weinstein domainshttps://projecteuclid.org/euclid.gt/1510859172<strong>Emmanuel Giroux</strong>, <strong>John Pardon</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 963--997.</p><p><strong>Abstract:</strong><br/>
We show that every Stein or Weinstein domain may be presented (up to deformation) as a Lefschetz fibration over the disk. The proof is an application of Donaldson’s quantitative transversality techniques.
</p>projecteuclid.org/euclid.gt/1510859172_20171116140618Thu, 16 Nov 2017 14:06 ESTThe codimension-one cohomology of $\mathrm{SL}_n \mathbb{Z}$https://projecteuclid.org/euclid.gt/1510859173<strong>Thomas Church</strong>, <strong>Andrew Putman</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 999--1032.</p><p><strong>Abstract:</strong><br/>
We prove that [math] , where [math] is the cohomological dimension of [math] , and similarly for [math] . We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group [math] . These theorems are derived from a presentation of the Steinberg module for [math] whose generators are integral apartment classes, generalizing Manin’s presentation for the Steinberg module of [math] . This presentation was originally constructed by Bykovskiĭ. We give a new topological proof of it.
</p>projecteuclid.org/euclid.gt/1510859173_20171116140618Thu, 16 Nov 2017 14:06 ESTA higher chromatic analogue of the image of $J$https://projecteuclid.org/euclid.gt/1510859174<strong>Craig Westerland</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 1033--1093.</p><p><strong>Abstract:</strong><br/>
We prove a higher chromatic analogue of Snaith’s theorem which identifies the [math] –theory spectrum as the localisation of the suspension spectrum of [math] away from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of [math] . Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the [math] –local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross–Hopkins dual of the sphere. Our main technical tool is a [math] –local notion generalising complex orientation to higher Eilenberg–MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture.
</p>projecteuclid.org/euclid.gt/1510859174_20171116140618Thu, 16 Nov 2017 14:06 ESTRational cohomology torihttps://projecteuclid.org/euclid.gt/1510859175<strong>Olivier Debarre</strong>, <strong>Zhi Jiang</strong>, <strong>Martí Lahoz</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 1095--1130.</p><p><strong>Abstract:</strong><br/> We study normal compact varieties in Fujiki’s class [math] whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on the degree of the Albanese morphism and the number of simple factors of the Albanese variety for rational cohomology tori of general type (hence projective) with rational singularities. Their properties are related to the birational geometry of smooth projective varieties of general type, maximal Albanese dimension, and with vanishing holomorphic Euler characteristic. We finish with the construction of series of examples. In an appendix, we show that there are no smooth rational cohomology tori of general type. The key technical ingredient is a result of Popa and Schnell on [math] –forms on smooth varieties of general type. </p>projecteuclid.org/euclid.gt/1510859175_20171116140618Thu, 16 Nov 2017 14:06 ESTOuter space for untwisted automorphisms of right-angled Artin groupshttps://projecteuclid.org/euclid.gt/1510859176<strong>Ruth Charney</strong>, <strong>Nathaniel Stambaugh</strong>, <strong>Karen Vogtmann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 1131--1178.</p><p><strong>Abstract:</strong><br/>
For a right-angled Artin group [math] , the untwisted outer automorphism group [math] is the subgroup of [math] generated by all of the Laurence–Servatius generators except twists (where a twist is an automorphism of the form [math] with [math] ). We define a space [math] on which [math] acts properly and prove that [math] is contractible, providing a geometric model for [math] and its subgroups. We also propose a geometric model for all of [math] , defined by allowing more general markings and metrics on points of [math] .
</p>projecteuclid.org/euclid.gt/1510859176_20171116140618Thu, 16 Nov 2017 14:06 ESTA very special EPW sextic and two IHS fourfoldshttps://projecteuclid.org/euclid.gt/1510859177<strong>Maria Donten-Bury</strong>, <strong>Bert van Geemen</strong>, <strong>Grzegorz Kapustka</strong>, <strong>Michał Kapustka</strong>, <strong>Jarosław Wiśniewski</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 1179--1230.</p><p><strong>Abstract:</strong><br/>
We show that the Hilbert scheme of two points on the Vinberg [math] surface has a two-to-one map onto a very symmetric EPW sextic [math] in [math] . The fourfold [math] is singular along [math] planes, [math] of which form a complete family of incident planes. This solves a problem of Morin and O’Grady and establishes that [math] is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer-type IHS fourfold [math] constructed by Donten-Bury and Wiśniewski [ On 81 symplectic resolutions of a 4–dimensional quotient by a group of order [math] , preprint (2014)]. We find that [math] is also related to the Debarre–Varley abelian fourfold.
</p>projecteuclid.org/euclid.gt/1510859177_20171116140618Thu, 16 Nov 2017 14:06 ESTArboreal singularitieshttps://projecteuclid.org/euclid.gt/1510859178<strong>David Nadler</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 1231--1274.</p><p><strong>Abstract:</strong><br/>
We introduce a class of combinatorial singularities of Lagrangian skeleta of symplectic manifolds. The link of each singularity is a finite regular cell complex homotopy equivalent to a bouquet of spheres. It is determined by its face poset, which is naturally constructed starting from a tree (nonempty finite acyclic graph). The choice of a root vertex of the tree leads to a natural front projection of the singularity along with an orientation of the edges of the tree. Microlocal sheaves along the singularity, calculated via the front projection, are equivalent to modules over the quiver given by the directed tree.
</p>projecteuclid.org/euclid.gt/1510859178_20171116140618Thu, 16 Nov 2017 14:06 ESTPresentation complexes with the fixed point propertyhttps://projecteuclid.org/euclid.gt/1510859179<strong>Iván Sadofschi Costa</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 2, 1275--1283.</p><p><strong>Abstract:</strong><br/>
We prove that there exists a compact two-dimensional polyhedron with the fixed point property and even Euler characteristic. This answers a question posed by R H Bing in 1969. We also settle a second question by Bing regarding the homotopy invariance of the fixed point property in low dimensions.
</p>projecteuclid.org/euclid.gt/1510859179_20171116140618Thu, 16 Nov 2017 14:06 ESTOn the topological contents of $\eta$–invariantshttps://projecteuclid.org/euclid.gt/1510859204<strong>Ulrich Bunke</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1285--1385.</p><p><strong>Abstract:</strong><br/>
We discuss a universal bordism invariant obtained from the Atiyah–Patodi–Singer [math] –invariant from the analytic and homotopy-theoretic point of view. Classical invariants like the Adams [math] –invariant, [math] –invariants and [math] –bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.
</p>projecteuclid.org/euclid.gt/1510859204_20171116140647Thu, 16 Nov 2017 14:06 ESTHomological stability for spaces of embedded surfaceshttps://projecteuclid.org/euclid.gt/1510859205<strong>Federico Cantero</strong>, <strong>Oscar Randal-Williams</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1387--1467.</p><p><strong>Abstract:</strong><br/> We study the space of oriented genus- [math] subsurfaces of a fixed manifold [math] and, in particular, its homological properties. We construct a “scanning map” which compares this space to the space of sections of a certain fibre bundle over [math] associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees. Our results are analogous to McDuff’s theorem on configuration spaces, extended from [math] –dimensional submanifolds to [math] –dimensional submanifolds. </p>projecteuclid.org/euclid.gt/1510859205_20171116140647Thu, 16 Nov 2017 14:06 ESTSutured Floer homology and invariants of Legendrian and transverse knotshttps://projecteuclid.org/euclid.gt/1510859206<strong>John Etnyre</strong>, <strong>David Vela-Vick</strong>, <strong>Rumen Zarev</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1469--1582.</p><p><strong>Abstract:</strong><br/>
Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus versions of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot [math] we associate distinguished classes [math] and [math] , which are each invariant under Legendrian or transverse isotopies of [math] . The distinguished class [math] is shown to agree with the Legendrian/transverse invariant defined by Lisca, Ozsváth, Stipsicz and Szabó despite a strikingly dissimilar definition. While our definitions and constructions only involve sutured Floer homology and contact geometry, the identification of our invariants with known invariants uses bordered sutured Floer homology to make explicit computations of maps between sutured Floer homology groups.
</p>projecteuclid.org/euclid.gt/1510859206_20171116140647Thu, 16 Nov 2017 14:06 ESTGenus-two trisections are standardhttps://projecteuclid.org/euclid.gt/1510859207<strong>Jeffrey Meier</strong>, <strong>Alexander Zupan</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1583--1630.</p><p><strong>Abstract:</strong><br/>
We show that the only closed [math] –manifolds admitting genus-two trisections are [math] and connected sums of [math] , [math] and [math] with two summands. Moreover, each of these manifolds admits a unique genus-two trisection up to diffeomorphism. The proof relies heavily on the combinatorics of genus-two Heegaard diagrams of [math] . As a corollary, we classify tunnel number one links with an integral cosmetic Dehn surgery.
</p>projecteuclid.org/euclid.gt/1510859207_20171116140647Thu, 16 Nov 2017 14:06 ESTThe higher Morita category of $\mathbb{E}_{n}$–algebrashttps://projecteuclid.org/euclid.gt/1510859208<strong>Rune Haugseng</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1631--1730.</p><p><strong>Abstract:</strong><br/>
We introduce simple models for associative algebras and bimodules in the context of nonsymmetric [math] –operads, and use these to construct an [math] –category of associative algebras, bimodules and bimodule homomorphisms in a monoidal [math] –category. By working with [math] –operads over [math] we iterate these definitions and generalize our construction to get an [math] –category of [math] –algebras and iterated bimodules in an [math] –monoidal [math] –category. Moreover, we show that if [math] is an [math] –monoidal [math] –category then the [math] –category of [math] –algebras in [math] has a natural [math] –monoidal structure. We also identify the mapping [math] –categories between two [math] –algebras, which allows us to define interesting nonconnective deloopings of the Brauer space of a commutative ring spectrum.
</p>projecteuclid.org/euclid.gt/1510859208_20171116140647Thu, 16 Nov 2017 14:06 ESTHierarchically hyperbolic spaces, I: Curve complexes for cubical groupshttps://projecteuclid.org/euclid.gt/1510859209<strong>Jason Behrstock</strong>, <strong>Mark Hagen</strong>, <strong>Alessandro Sisto</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1731--1804.</p><p><strong>Abstract:</strong><br/> In the context of [math] cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a factor system , and the role of the curve graph is played by the contact graph . There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur–Minsky-style distance formula. We then define a hierarchically hyperbolic space ; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichmüller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock and Minsky, of Eskin, Masur and Rafi, of Hamenstädt, and of Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch. </p>projecteuclid.org/euclid.gt/1510859209_20171116140647Thu, 16 Nov 2017 14:06 ESTStrong accessibility for finitely presented groupshttps://projecteuclid.org/euclid.gt/1510859210<strong>Larsen Louder</strong>, <strong>Nicholas Touikan</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1805--1835.</p><p><strong>Abstract:</strong><br/> A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost) finitely presented groups and show that it is finite, provided the group in question doesn’t contain any slender subgroups with infinite dihedral quotients and satisfies an ascending chain condition on certain chains of subgroups of edge groups. As a corollary, slender JSJ hierarchies of finitely presented subgroups of [math] or of hyperbolic groups which are (virtually) without [math] –torsion are finite. </p>projecteuclid.org/euclid.gt/1510859210_20171116140647Thu, 16 Nov 2017 14:06 ESTBuilding Anosov flows on $3$–manifoldshttps://projecteuclid.org/euclid.gt/1510859211<strong>François Béguin</strong>, <strong>Christian Bonatti</strong>, <strong>Bin Yu</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 3, 1837--1930.</p><p><strong>Abstract:</strong><br/> We prove we can build (transitive or nontransitive) Anosov flows on closed three-dimensional manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications of this result; for example: We build a closed three-dimensional manifold supporting both a transitive Anosov vector field and a nontransitive Anosov vector field. For any [math] , we build a closed three-dimensional manifold [math] supporting at least [math] pairwise different Anosov vector fields. We build transitive hyperbolic attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive hyperbolic attractors. We build a transitive Anosov vector field admitting infinitely many pairwise nonisotopic transverse tori. </p>projecteuclid.org/euclid.gt/1510859211_20171116140647Thu, 16 Nov 2017 14:06 ESTGrowth and order of automorphisms of free groups and free Burnside groupshttps://projecteuclid.org/euclid.gt/1508437635<strong>Rémi Coulon</strong>, <strong>Arnaud Hilion</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 4, 1969--2014.</p><p><strong>Abstract:</strong><br/>
We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.
</p>projecteuclid.org/euclid.gt/1508437635_20171116140727Thu, 16 Nov 2017 14:07 ESTA geometric construction of colored HOMFLYPT homologyhttps://projecteuclid.org/euclid.gt/1510859271<strong>Benjamin Webster</strong>, <strong>Geordie Williamson</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2557--2600.</p><p><strong>Abstract:</strong><br/> The aim of this paper is twofold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov and Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors’ previous work on Soergel bimodules. All the differentials and gradings which appear in the construction of HOMFLYPT homology are given a geometric interpretation. In fact, with only minor modifications, we can extend this construction to give a categorification of the colored HOMFLYPT polynomial, colored HOMFLYPT homology . We show that it is in fact a knot invariant categorifying the colored HOMFLYPT polynomial and that it coincides with the categorification proposed by Mackaay, Stošić and Vaz. </p>projecteuclid.org/euclid.gt/1510859271_20171116140818Thu, 16 Nov 2017 14:08 ESTCategorical cell decomposition of quantized symplectic algebraic varietieshttps://projecteuclid.org/euclid.gt/1510859272<strong>Gwyn Bellamy</strong>, <strong>Christopher Dodd</strong>, <strong>Kevin McGerty</strong>, <strong>Thomas Nevins</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2601--2681.</p><p><strong>Abstract:</strong><br/>
We prove a new symplectic analogue of Kashiwara’s equivalence from [math] –module theory. As a consequence, we establish a structure theory for module categories over deformation-quantizations that mirrors, at a higher categorical level, the Białynicki-Birula stratification of a variety with an action of the multiplicative group [math] . The resulting categorical cell decomposition provides an algebrogeometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as [math] –theory and Hochschild homology of module categories of interest in geometric representation theory.
</p>projecteuclid.org/euclid.gt/1510859272_20171116140818Thu, 16 Nov 2017 14:08 ESTThe nonuniqueness of the tangent cones at infinity of Ricci-flat manifoldshttps://projecteuclid.org/euclid.gt/1510859273<strong>Kota Hattori</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2683--2723.</p><p><strong>Abstract:</strong><br/>
Colding and Minicozzi established the uniqueness of the tangent cones at infinity of Ricci-flat manifolds with Euclidean volume growth where at least one tangent cone at infinity has a smooth cross section. In this paper, we raise an example of a Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, we construct a complete Ricci-flat manifold of dimension [math] with non-Euclidean volume growth that has infinitely many tangent cones at infinity where one of them has a smooth cross section.
</p>projecteuclid.org/euclid.gt/1510859273_20171116140818Thu, 16 Nov 2017 14:08 ESTSmooth Kuranishi atlases with isotropyhttps://projecteuclid.org/euclid.gt/1510859274<strong>Dusa McDuff</strong>, <strong>Katrin Wehrheim</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2725--2809.</p><p><strong>Abstract:</strong><br/> Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite-dimensional reductions, given by smooth sections that are equivariant under a finite isotropy group. Building on our notions of topological Kuranishi atlases and perturbation constructions in the case of trivial isotropy, we develop a theory of Kuranishi atlases and cobordisms that transparently resolves the challenges posed by nontrivial isotropy. We assign to a cobordism class of weak Kuranishi atlases both a virtual moduli cycle (a cobordism class of weighted branched manifolds) and a virtual fundamental class (a Čech homology class). </p>projecteuclid.org/euclid.gt/1510859274_20171116140818Thu, 16 Nov 2017 14:08 ESTBrown's moduli spaces of curves and the gravity operadhttps://projecteuclid.org/euclid.gt/1510859275<strong>Clément Dupont</strong>, <strong>Bruno Vallette</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2811--2850.</p><p><strong>Abstract:</strong><br/>
This paper is built on the following observation: the purity of the mixed Hodge structure on the cohomology of Brown’s moduli spaces is essentially equivalent to the freeness of the dihedral operad underlying the gravity operad. We prove these two facts by relying on both the geometric and the algebraic aspects of the problem: the complete geometric description of the cohomology of Brown’s moduli spaces and the coradical filtration of cofree cooperads. This gives a conceptual proof of an identity of Bergström and Brown which expresses the Betti numbers of Brown’s moduli spaces via the inversion of a generating series. This also generalizes the Salvatore–Tauraso theorem on the nonsymmetric Lie operad.
</p>projecteuclid.org/euclid.gt/1510859275_20171116140818Thu, 16 Nov 2017 14:08 ESTOn the second homology group of the Torelli subgroup of $\mathrm{Aut}(F_n)$https://projecteuclid.org/euclid.gt/1510859276<strong>Matthew Day</strong>, <strong>Andrew Putman</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2851--2896.</p><p><strong>Abstract:</strong><br/>
Let [math] be the Torelli subgroup of [math] . We give an explicit finite set of generators for [math] as a [math] –module. Corollaries include a version of surjective representation stability for [math] , the vanishing of the [math] –coinvariants of [math] , and the vanishing of the second rational homology group of the level [math] congruence subgroup of [math] . Our generating set is derived from a new group presentation for [math] which is infinite but which has a simple recursive form.
</p>projecteuclid.org/euclid.gt/1510859276_20171116140818Thu, 16 Nov 2017 14:08 ESTTautological integrals on curvilinear Hilbert schemeshttps://projecteuclid.org/euclid.gt/1510859277<strong>Gergely Bérczi</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2897--2944.</p><p><strong>Abstract:</strong><br/>
We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety [math] as a projective completion of the nonreductive quotient of holomorphic map germs from the complex line into [math] by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes.
</p>projecteuclid.org/euclid.gt/1510859277_20171116140818Thu, 16 Nov 2017 14:08 ESTConvexity of the extended K-energy and the large time behavior of the weak Calabi flowhttps://projecteuclid.org/euclid.gt/1510859278<strong>Robert Berman</strong>, <strong>Tamás Darvas</strong>, <strong>Chinh Lu</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2945--2988.</p><p><strong>Abstract:</strong><br/>
Let [math] be a compact connected Kähler manifold and denote by [math] the metric completion of the space of Kähler potentials [math] with respect to the [math] –type path length metric [math] . First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to [math] is a [math] –lsc functional that is convex along finite-energy geodesics. Second, following the program of J Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space [math] . This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the [math] –metric or it [math] –converges to some minimizer of the K-energy inside [math] . This gives the first concrete result about the long-time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. We investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. Finally, when a cscK metric exists in [math] , our results imply that the weak Calabi flow [math] –converges to such a metric.
</p>projecteuclid.org/euclid.gt/1510859278_20171116140818Thu, 16 Nov 2017 14:08 ESTOn $5$–manifolds with free fundamental group and simple boundary links in $S^5$https://projecteuclid.org/euclid.gt/1510859279<strong>Matthias Kreck</strong>, <strong>Yang Su</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 2989--3008.</p><p><strong>Abstract:</strong><br/>
We classify compact oriented [math] –manifolds with free fundamental group and [math] a torsion-free abelian group in terms of the second homotopy group considered as a [math] –module, the cup product on the second cohomology of the universal covering, and the second Stiefel–Whitney class of the universal covering. We apply this to the classification of simple boundary links of [math] –spheres in [math] . Using this we give a complete algebraic picture of closed [math] –manifolds with free fundamental group and trivial second homology group.
</p>projecteuclid.org/euclid.gt/1510859279_20171116140818Thu, 16 Nov 2017 14:08 ESTOn the Fano variety of linear spaces contained in two odd-dimensional quadricshttps://projecteuclid.org/euclid.gt/1510859280<strong>Carolina Araujo</strong>, <strong>Cinzia Casagrande</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 3009--3045.</p><p><strong>Abstract:</strong><br/> We describe the geometry of the [math] –dimensional Fano manifold [math] parametrizing [math] –planes in a smooth complete intersection [math] of two quadric hypersurfaces in the complex projective space [math] for [math] . We show that there are exactly [math] distinct isomorphisms in codimension one between [math] and the blow-up of [math] at [math] general points, parametrized by the [math] distinct [math] –planes contained in [math] , and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of [math] , as well as their dual cones of curves. Finally, we determine the automorphism group of [math] . These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces ( [math] ). </p>projecteuclid.org/euclid.gt/1510859280_20171116140818Thu, 16 Nov 2017 14:08 ESTStable homology of surface diffeomorphism groups made discretehttps://projecteuclid.org/euclid.gt/1510859281<strong>Sam Nariman</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 3047--3092.</p><p><strong>Abstract:</strong><br/>
We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that [math] –diffeomorphisms of surfaces as family of discrete groups exhibit homological stability . We show that the stable homology of [math] –diffeomorphisms of surfaces as discrete groups is the same as homology of certain infinite loop space related to Haefliger’s classifying space of foliations of codimension [math] . We use this infinite loop space to obtain new results about (non)triviality of characteristic classes of flat surface bundles and codimension- [math] foliations.
</p>projecteuclid.org/euclid.gt/1510859281_20171116140818Thu, 16 Nov 2017 14:08 ESTKato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemeshttps://projecteuclid.org/euclid.gt/1510859284<strong>David Carchedi</strong>, <strong>Sarah Scherotzke</strong>, <strong>Nicolò Sibilla</strong>, <strong>Mattia Talpo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 3093--3158.</p><p><strong>Abstract:</strong><br/>
For a log scheme locally of finite type over [math] , a natural candidate for its profinite homotopy type is the profinite completion of its Kato–Nakayama space. Alternatively, one may consider the profinite homotopy type of the underlying topological stack of its infinite root stack. Finally, for a log scheme not necessarily over [math] , another natural candidate is the profinite étale homotopy type of its infinite root stack. We prove that, for a fine saturated log scheme locally of finite type over [math] , these three notions agree. In particular, we construct a comparison map from the Kato–Nakayama space to the underlying topological stack of the infinite root stack, and prove that it induces an equivalence on profinite completions. In light of these results, we define the profinite homotopy type of a general fine saturated log scheme as the profinite étale homotopy type of its infinite root stack.
</p>projecteuclid.org/euclid.gt/1510859284_20171116140818Thu, 16 Nov 2017 14:08 ESTPositive simplicial volume implies virtually positive Seifert volume for $3$–manifoldshttps://projecteuclid.org/euclid.gt/1510859285<strong>Pierre Derbez</strong>, <strong>Yi Liu</strong>, <strong>Hongbin Sun</strong>, <strong>Shicheng Wang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 5, 3159--3190.</p><p><strong>Abstract:</strong><br/>
We show that for any closed orientable [math] –manifold with positive simplicial volume, the growth of the Seifert volume of its finite covers is faster than the linear rate. In particular, each closed orientable [math] –manifold with positive simplicial volume has virtually positive Seifert volume. The result reveals certain fundamental differences between the representation volumes of hyperbolic type and Seifert type. The proof is based on developments and interactions of recent results on virtual domination and on virtual representation volumes of [math] –manifolds.
</p>projecteuclid.org/euclid.gt/1510859285_20171116140818Thu, 16 Nov 2017 14:08 ESTIndependence of satellites of torus knots in the smooth concordance grouphttps://projecteuclid.org/euclid.gt/1510859319<strong>Juanita Pinzón-Caicedo</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3191--3211.</p><p><strong>Abstract:</strong><br/>
The main goal of this article is to obtain a condition under which an infinite collection [math] of satellite knots (with companion a positive torus knot and pattern similar to the Whitehead link) freely generates a subgroup of infinite rank in the smooth concordance group. This goal is attained by examining both the instanton moduli space over a [math] –manifold with tubular ends and the corresponding Chern–Simons invariant of the adequate [math] –dimensional portion of the [math] –manifold. More specifically, the result is derived from Furuta’s criterion for the independence of Seifert fibred homology spheres in the homology cobordism group of oriented homology [math] –spheres. Indeed, we first associate to [math] the corresponding collection of [math] –fold covers of the [math] –sphere branched over the elements of [math] and then introduce definite cobordisms from the aforementioned covers of the satellites to a number of Seifert fibered homology spheres. This allows us to apply Furuta’s criterion and thus obtain a condition that guarantees the independence of the family [math] in the smooth concordance group.
</p>projecteuclid.org/euclid.gt/1510859319_20171116140852Thu, 16 Nov 2017 14:08 ESTThe chromatic splitting conjecture at $n=p=2$https://projecteuclid.org/euclid.gt/1510859320<strong>Agnès Beaudry</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3213--3230.</p><p><strong>Abstract:</strong><br/>
We show that the strongest form of Hopkins’ chromatic splitting conjecture, as stated by Hovey, cannot hold at chromatic level [math] at the prime [math] . More precisely, for [math] , the mod [math] Moore spectrum, we prove that [math] is not zero when [math] is congruent to [math] modulo [math] . We explain how this contradicts the decomposition of [math] predicted by the chromatic splitting conjecture.
</p>projecteuclid.org/euclid.gt/1510859320_20171116140852Thu, 16 Nov 2017 14:08 ESTVirtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-foldshttps://projecteuclid.org/euclid.gt/1510859321<strong>Dennis Borisov</strong>, <strong>Dominic Joyce</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3231--3311.</p><p><strong>Abstract:</strong><br/> Let [math] be a separated, [math] –shifted symplectic derived [math] –scheme, in the sense of Pantev, Toën, Vezzosi and Vaquié (2013), of complex virtual dimension [math] , and [math] the underlying complex analytic topological space. We prove that [math] can be given the structure of a derived smooth manifold [math] , of real virtual dimension [math] . This [math] is not canonical, but is independent of choices up to bordisms fixing the underlying topological space [math] . There is a one-to-one correspondence between orientations on [math] and orientations on [math] . Because compact, oriented derived manifolds have virtual classes, this means that proper, oriented [math] –shifted symplectic derived [math] –schemes have virtual classes, in either homology or bordism. This is surprising, as conventional algebrogeometric virtual cycle methods fail in this case. Our virtual classes have half the expected dimension. Now derived moduli schemes of coherent sheaves on a Calabi–Yau [math] –fold are expected to be [math] –shifted symplectic (this holds for stacks). We propose to use our virtual classes to define new Donaldson–Thomas style invariants “counting” (semi)stable coherent sheaves on Calabi–Yau [math] –folds [math] over [math] , which should be unchanged under deformations of [math] . </p>projecteuclid.org/euclid.gt/1510859321_20171116140852Thu, 16 Nov 2017 14:08 ESTKoszul duality patterns in Floer theoryhttps://projecteuclid.org/euclid.gt/1510859322<strong>Tolga Etgü</strong>, <strong>Yankı Lekili</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3313--3389.</p><p><strong>Abstract:</strong><br/>
We study symplectic invariants of the open symplectic manifolds [math] obtained by plumbing cotangent bundles of 2–spheres according to a plumbing tree [math] . For any tree [math] , we calculate (DG) algebra models of the Fukaya category [math] of closed exact Lagrangians in [math] and the wrapped Fukaya category [math] . When [math] is a Dynkin tree of type [math] or [math] (and conjecturally also for [math] ), we prove that these models for the Fukaya category [math] and [math] are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of [math] for [math] , based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.
</p>projecteuclid.org/euclid.gt/1510859322_20171116140852Thu, 16 Nov 2017 14:08 ESTA complex hyperbolic Riley slicehttps://projecteuclid.org/euclid.gt/1510859323<strong>John Parker</strong>, <strong>Pierre Will</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3391--3451.</p><p><strong>Abstract:</strong><br/>
We study subgroups of [math] generated by two noncommuting unipotent maps [math] and [math] whose product [math] is also unipotent. We call [math] the set of conjugacy classes of such groups. We provide a set of coordinates on [math] that make it homeomorphic to [math] . By considering the action on complex hyperbolic space [math] of groups in [math] , we describe a two-dimensional disc [math] in [math] that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for [math] –triangle groups. We also consider a particular group on the boundary of the disc [math] where the commutator [math] is also unipotent. We show that the boundary of the quotient orbifold associated to the latter group gives a spherical CR uniformisation of the Whitehead link complement.
</p>projecteuclid.org/euclid.gt/1510859323_20171116140852Thu, 16 Nov 2017 14:08 ESTThe nilpotence theorem for the algebraic $K$–theory of the sphere spectrumhttps://projecteuclid.org/euclid.gt/1510859324<strong>Andrew Blumberg</strong>, <strong>Michael Mandell</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3453--3466.</p><p><strong>Abstract:</strong><br/>
We prove that in the graded commutative ring [math] , all positive degree elements are multiplicatively nilpotent. The analogous statements also hold for [math] and [math] .
</p>projecteuclid.org/euclid.gt/1510859324_20171116140852Thu, 16 Nov 2017 14:08 ESTQuasi-isometric classification of right-angled Artin groups, I: The finite out casehttps://projecteuclid.org/euclid.gt/1510859325<strong>Jingyin Huang</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3467--3537.</p><p><strong>Abstract:</strong><br/>
Let [math] and [math] be two right-angled Artin groups. We show they are quasi-isometric if and only if they are isomorphic, under the assumption that the outer automorphism groups [math] and [math] are finite. If we only assume [math] is finite, then [math] is quasi-isometric to [math] if and only if [math] is isomorphic to a subgroup of finite index in [math] . In this case, we give an algorithm to determine whether [math] and [math] are quasi-isometric by looking at their defining graphs.
</p>projecteuclid.org/euclid.gt/1510859325_20171116140852Thu, 16 Nov 2017 14:08 ESTMaximal representations, non-Archimedean Siegel spaces, and buildingshttps://projecteuclid.org/euclid.gt/1510859326<strong>Marc Burger</strong>, <strong>Maria Beatrice Pozzetti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3539--3599.</p><p><strong>Abstract:</strong><br/>
Let [math] be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in [math] . We show that ultralimits of maximal representations in [math] admit such a framing, and that all maximal framed representations satisfy a suitable generalization of the classical collar lemma. In particular, this establishes a collar lemma for all maximal representations into [math] . We then describe a procedure to get from representations in [math] interesting actions on affine buildings, and in the case of representations admitting a maximal framing, we describe the structure of the elements of the group acting with zero translation length.
</p>projecteuclid.org/euclid.gt/1510859326_20171116140852Thu, 16 Nov 2017 14:08 EST$C^0$ approximations of foliationshttps://projecteuclid.org/euclid.gt/1510859327<strong>William Kazez</strong>, <strong>Rachel Roberts</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3601--3657.</p><p><strong>Abstract:</strong><br/>
Suppose that [math] is a transversely oriented, codimension-one foliation of a connected, closed, oriented [math] –manifold. Suppose also that [math] has continuous tangent plane field and is taut ; that is, closed smooth transversals to [math] pass through every point of [math] . We show that if [math] is not the product foliation [math] , then [math] can be [math] approximated by weakly symplectically fillable, universally tight contact structures. This extends work of Eliashberg and Thurston on approximations of taut, transversely oriented [math] foliations to the class of foliations that often arise in branched surface constructions of foliations. This allows applications of contact topology and Floer theory beyond the category of [math] foliated spaces.
</p>projecteuclid.org/euclid.gt/1510859327_20171116140852Thu, 16 Nov 2017 14:08 ESTBoundaries and automorphisms of hierarchically hyperbolic spaceshttps://projecteuclid.org/euclid.gt/1510859328<strong>Matthew Durham</strong>, <strong>Mark Hagen</strong>, <strong>Alessandro Sisto</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3659--3758.</p><p><strong>Abstract:</strong><br/> Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite-type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space [math] , we build a bordification of [math] compatible with its hierarchically hyperbolic structure. If [math] is proper, eg a hierarchically hyperbolic group such as the mapping class group, we get a compactification of [math] ; we also prove that our construction generalizes the Gromov boundary of a hyperbolic space. In our first main set of applications, we introduce a notion of geometrical finiteness for hierarchically hyperbolic subgroups of hierarchically hyperbolic groups in terms of boundary embeddings. As primary examples of geometrical finiteness, we prove that the natural inclusions of finitely generated Veech groups and the Leininger–Reid combination subgroups extend to continuous embeddings of their Gromov boundaries into the boundary of the mapping class group, both of which fail to happen with the Thurston compactification of Teichmüller space. Our second main set of applications are dynamical and structural, built upon our classification of automorphisms of hierarchically hyperbolic spaces and analysis of how the various types of automorphisms act on the boundary. We prove a generalization of the Handel–Mosher “omnibus subgroup theorem” for mapping class groups to all hierarchically hyperbolic groups, obtain a new proof of the Caprace–Sageev rank-rigidity theorem for many [math] cube complexes, and identify the boundary of a hierarchically hyperbolic group as its Poisson boundary; these results rely on a theorem detecting irreducible axial elements of a group acting on a hierarchically hyperbolic space (which generalize pseudo-Anosov elements of the mapping class group and rank-one isometries of a cube complex not virtually stabilizing a hyperplane). </p>projecteuclid.org/euclid.gt/1510859328_20171116140852Thu, 16 Nov 2017 14:08 ESTThurston norm via Fox calculushttps://projecteuclid.org/euclid.gt/1510859329<strong>Stefan Friedl</strong>, <strong>Kevin Schreve</strong>, <strong>Stephan Tillmann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3759--3784.</p><p><strong>Abstract:</strong><br/>
In 1976 Thurston associated to a [math] –manifold [math] a marked polytope in [math] , which measures the minimal complexity of surfaces representing homology classes and determines all fibered classes in [math] . Recently the first and third authors associated to a presentation [math] with two generators and one relator a marked polytope in [math] and showed that it determines the Bieri–Neumann–Strebel invariant of [math] . We show that if the fundamental group of a [math] –manifold [math] admits such a presentation [math] , then the corresponding marked polytopes in [math] agree.
</p>projecteuclid.org/euclid.gt/1510859329_20171116140852Thu, 16 Nov 2017 14:08 ESTThe $L^p$–diameter of the group of area-preserving diffeomorphisms of $S^2$https://projecteuclid.org/euclid.gt/1510859330<strong>Michael Brandenbursky</strong>, <strong>Egor Shelukhin</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 21, Number 6, 3785--3810.</p><p><strong>Abstract:</strong><br/>
We show that for each [math] , the [math] –metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasimorphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map [math] from the configuration space of [math] points on [math] to the moduli space of complex rational curves with [math] marked points.
</p>projecteuclid.org/euclid.gt/1510859330_20171116140852Thu, 16 Nov 2017 14:08 ESTDe Rham theory of exploded manifoldshttps://projecteuclid.org/euclid.gt/1513774909<strong>Brett Parker</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 1--54.</p><p><strong>Abstract:</strong><br/>
This paper extends de Rham theory of smooth manifolds to exploded manifolds. Included are versions of Stokes’ theorem, de Rham cohomology, Poincaré duality, and integration along the fiber. The resulting de Rham cohomology theory of exploded manifolds is used in a separate paper (arXiv:1102.0158) to define Gromov–Witten invariants of exploded manifolds.
</p>projecteuclid.org/euclid.gt/1513774909_20171220080154Wed, 20 Dec 2017 08:01 ESTGauge-reversing maps on cones, and Hilbert and Thompson isometrieshttps://projecteuclid.org/euclid.gt/1513774910<strong>Cormac Walsh</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 55--104.</p><p><strong>Abstract:</strong><br/>
We show that a cone admits a gauge-reversing map if and only if it is a symmetric cone. We use this to prove that every isometry of a Hilbert geometry is a projectivity unless the Hilbert geometry is the projective space of a non-Lorentzian symmetric cone, in which case the projectivity group is of index two in the isometry group. We also determine the isometry group of the Thompson geometry on a cone.
</p>projecteuclid.org/euclid.gt/1513774910_20171220080154Wed, 20 Dec 2017 08:01 ESTCentral limit theorems for mapping class groups and $\mathrm{Out}(F_N)$https://projecteuclid.org/euclid.gt/1513774911<strong>Camille Horbez</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 105--156.</p><p><strong>Abstract:</strong><br/>
We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on [math] , each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric on outer space). In the mapping class group case, this describes the spread of the hyperbolic length of a simple closed curve on the surface after applying a random product of mapping classes. In the case of [math] , this describes the spread of the length of primitive conjugacy classes in [math] under random products of outer automorphisms. Both results are based on a general criterion for establishing a central limit theorem for the Busemann cocycle on the horoboundary of a metric space, applied to either the Teichmüller space of the surface or to the Culler–Vogtmann outer space.
</p>projecteuclid.org/euclid.gt/1513774911_20171220080154Wed, 20 Dec 2017 08:01 ESTDynamics on flag manifolds: domains of proper discontinuity and cocompactnesshttps://projecteuclid.org/euclid.gt/1513774912<strong>Michael Kapovich</strong>, <strong>Bernhard Leeb</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 157--234.</p><p><strong>Abstract:</strong><br/>
For noncompact semisimple Lie groups [math] with finite center, we study the dynamics of the actions of their discrete subgroups [math] on the associated partial flag manifolds [math] . Our study is based on the observation, already made in the deep work of Benoist, that they exhibit also in higher rank a certain form of convergence-type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the [math] –action on various domains of proper discontinuity, in particular on domains in the full flag manifold [math] . In the regular case (eg of [math] –Anosov subgroups), we prove the nonemptiness of such domains if [math] has (locally) at least one noncompact simple factor not of the type [math] , [math] or [math] by showing the nonexistence of certain ball packings of the visual boundary.
</p>projecteuclid.org/euclid.gt/1513774912_20171220080154Wed, 20 Dec 2017 08:01 ESTA mathematical theory of the gauged linear sigma modelhttps://projecteuclid.org/euclid.gt/1513774913<strong>Huijun Fan</strong>, <strong>Tyler Jarvis</strong>, <strong>Yongbin Ruan</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 235--303.</p><p><strong>Abstract:</strong><br/>
We construct a mathematical theory of Witten’s Gauged Linear Sigma Model (GLSM). Our theory applies to a wide range of examples, including many cases with nonabelian gauge group.
Both the Gromov–Witten theory of a Calabi–Yau complete intersection [math] and the Landau–Ginzburg dual (FJRW theory) of [math] can be expressed as gauged linear sigma models. Furthermore, the Landau–Ginzburg/Calabi–Yau correspondence can be interpreted as a variation of the moment map or a deformation of GIT in the GLSM. This paper focuses primarily on the algebraic theory, while a companion article will treat the analytic theory.
</p>projecteuclid.org/euclid.gt/1513774913_20171220080154Wed, 20 Dec 2017 08:01 ESTChord arc properties for constant mean curvature diskshttps://projecteuclid.org/euclid.gt/1513774914<strong>William Meeks</strong>, <strong>Giuseppe Tinaglia</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 305--322.</p><p><strong>Abstract:</strong><br/>
We prove a chord arc type bound for disks embedded in [math] with constant mean curvature that does not depend on the value of the mean curvature. This bound is inspired by and generalizes the weak chord arc bound of Colding and Minicozzi in Proposition 2.1 of Ann. of Math. 167 (2008) 211–243 for embedded minimal disks. Like in the minimal case, this chord arc bound is a fundamental tool for studying complete constant mean curvature surfaces embedded in [math] with finite topology or with positive injectivity radius.
</p>projecteuclid.org/euclid.gt/1513774914_20171220080154Wed, 20 Dec 2017 08:01 ESTGromov–Witten invariants of the Hilbert schemes of points of a K3 surfacehttps://projecteuclid.org/euclid.gt/1513774915<strong>Georg Oberdieck</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 323--437.</p><p><strong>Abstract:</strong><br/>
We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.
Let [math] be a K3 surface and let [math] be the Hilbert scheme of [math] points of [math] . In the case of elliptically fibered K3 surfaces [math] , we calculate genus-0 Gromov–Witten invariants of [math] , which count rational curves incident to two generic fibers of the induced Lagrangian fibration [math] . The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form.
We also prove results for genus-0 Gromov–Witten invariants of [math] for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov–Witten invariants of the Hilbert scheme of two points of [math] , where [math] is an elliptic curve.
Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on [math] with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of [math] . We prove the conjecture in the first nontrivial case [math] . As a corollary, we find that the full genus-0 Gromov–Witten theory of [math] in primitive classes is governed by Jacobi forms.
We present two applications. A conjecture relating genus-1 invariants of [math] to the Igusa cusp form was proposed in joint work with R Pandharipande. Our results prove the conjecture when [math] . Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through two general points.
</p>projecteuclid.org/euclid.gt/1513774915_20171220080154Wed, 20 Dec 2017 08:01 ESTDetecting sphere boundaries of hyperbolic groupshttps://projecteuclid.org/euclid.gt/1513774916<strong>Benjamin Beeker</strong>, <strong>Nir Lazarovich</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 439--470.</p><p><strong>Abstract:</strong><br/>
We show that a one-ended simply connected at infinity hyperbolic group [math] with enough codimension- [math] surface subgroups has [math] . By work of Markovic (2013), our result gives a new characterization of virtually fundamental groups of hyperbolic [math] –manifolds.
</p>projecteuclid.org/euclid.gt/1513774916_20171220080154Wed, 20 Dec 2017 08:01 ESTEquivariant characteristic classes of external and symmetric products of varietieshttps://projecteuclid.org/euclid.gt/1513774917<strong>Laurenţiu Maxim</strong>, <strong>Jörg Schürmann</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 471--515.</p><p><strong>Abstract:</strong><br/>
We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasiprojective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel–Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.
</p>projecteuclid.org/euclid.gt/1513774917_20171220080154Wed, 20 Dec 2017 08:01 ESTHyperbolic extensions of free groupshttps://projecteuclid.org/euclid.gt/1513774918<strong>Spencer Dowdall</strong>, <strong>Samuel Taylor</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 517--570.</p><p><strong>Abstract:</strong><br/>
Given a finitely generated subgroup [math] of the outer automorphism group of the rank- [math] free group [math] , there is a corresponding free group extension [math] . We give sufficient conditions for when the extension [math] is hyperbolic. In particular, we show that if all infinite-order elements of [math] are atoroidal and the action of [math] on the free factor complex of [math] has a quasi-isometric orbit map, then [math] is hyperbolic. As an application, we produce examples of hyperbolic [math] –extensions [math] for which [math] has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex. This may be of independent interest.
</p>projecteuclid.org/euclid.gt/1513774918_20171220080154Wed, 20 Dec 2017 08:01 ESTComplete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$https://projecteuclid.org/euclid.gt/1513774919<strong>Antonio Alarcón</strong>, <strong>Ildefonso Castro-Infantes</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 571--590.</p><p><strong>Abstract:</strong><br/>
In this paper we prove that, given an open Riemann surface [math] and an integer [math] , the set of complete conformal minimal immersions [math] with [math] forms a dense subset in the space of all conformal minimal immersions [math] endowed with the compact-open topology. Moreover, we show that every domain in [math] contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface.
Our method of proof can be adapted to give analogous results for nonorientable minimal surfaces in [math] , complex curves in [math] , holomorphic null curves in [math] , and holomorphic Legendrian curves in [math] .
</p>projecteuclid.org/euclid.gt/1513774919_20171220080154Wed, 20 Dec 2017 08:01 ESTIntrinsic structure of minimal discs in metric spaceshttps://projecteuclid.org/euclid.gt/1513774920<strong>Alexander Lytchak</strong>, <strong>Stefan Wenger</strong>. <p><strong>Source: </strong>Geometry & Topology, Volume 22, Number 1, 591--644.</p><p><strong>Abstract:</strong><br/>
We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces.
</p>projecteuclid.org/euclid.gt/1513774920_20171220080154Wed, 20 Dec 2017 08:01 EST