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Compacta and are said to admit a stable intersection in if there are maps and such that for every sufficiently close continuous approximations and of and , we have . The unstable intersection conjecture asserts that and do not admit a stable intersection in if and only if . This conjecture was intensively studied and confirmed in many cases. we prove the unstable intersection conjecture in all the remaining cases except the case , and , which still remains open.
We give a geometric interpretation of the maximal Satake compactification of symmetric spaces of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable –invariant Finsler metric on . As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces for arbitrary discrete subgroups . These bordifications result from attaching –quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion-free case, to a question of Haïssinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.
We investigate Friedl and Lück’s universal –torsion for descending HNN extensions of finitely generated free groups, and so in particular for -by- groups. This invariant induces a seminorm on the first cohomology of the group which is an analogue of the Thurston norm for –manifold groups.
We prove that this Thurston seminorm is an upper bound for the Alexander seminorm defined by McMullen, as well as for the higher Alexander seminorms defined by Harvey. The same inequalities are known to hold for –manifold groups.
We also prove that the Newton polytopes of the universal –torsion of a descending HNN extension of locally determine the Bieri–Neumann–Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri–Neumann–Strebel invariant of a descending HNN extension of has finitely many connected components.
When the HNN extension is taken over along a polynomially growing automorphism with unipotent image in , we show that the Newton polytope of the universal –torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations and the Thurston norm all coincide.
A line arrangement of lines in satisfies the Hirzebruch property if each line intersect others in points. Hirzebruch asked in 1985 if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in is real, confirming that there exist exactly four such arrangements.
A once extended –dimensional topological field theory is a symmetric monoidal functor (taking values in a chosen target symmetric monoidal –category) assigning values to –manifolds, –manifolds, and –manifolds. We show that if is at least once extended and the value assigned to the –torus is invertible, then the entire topological field theory is invertible, that is, it factors through the maximal Picard –category of the target. Similar results are shown to hold in the presence of arbitrary tangential structures.
When is a convex-cocompact action of a discrete group on a noncompact rank-one symmetric space , there is a natural lower bound for the Hausdorff dimension of the limit set , given by the Ahlfors regular conformal dimension of . We show that equality is achieved precisely when stabilizes an isometric copy of some noncompact rank-one symmetric space in on which it acts with compact quotient. This generalizes a theorem of Bonk and Kleiner, who proved it in the case that is real hyperbolic.
To prove our main theorem, we study tangents of Lipschitz differentiability spaces that are embedded in a Carnot group . We show that almost all tangents are isometric to a Carnot subgroup of , at least when they are rectifiably connected. This extends a theorem of Cheeger, who proved it for PI spaces that are embedded in Euclidean space.
We study the possibility of realizing exotic smooth structures on finitely punctured simply connected closed –manifolds as leaves of a codimension-one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open –manifolds which are not diffeomorphic to any leaf of a codimension-one foliation on a compact manifold. These examples include some exotic ’s and exotic cylinders .
We prove a Smith-type inequality for regular covering spaces in monopole Floer homology. Using the monopole Floer/Heegaard Floer correspondence, we deduce that if a –manifold admits a –sheeted regular cover that is a ––space (for prime), then is a ––space. Further, we obtain constraints on surgeries on a knot being regular covers over other surgeries on the same knot, and over surgeries on other knots.
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most , , or polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main results are explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these formulas, we build on work of Thurston, who showed that the convex triangulations correspond to orbits of vectors of positive norm in a Hermitian lattice . First, we extend this result to convex square and hexagon tilings. Then, we explicitly compute the relevant lattice . Next, we integrate the Siegel theta function for to produce a modular form whose Fourier coefficients encode the weighted number of tilings. Finally, we determine the formulas using finite-dimensionality of spaces of modular forms.
We study the Seiberg–Witten invariant of smooth spin –manifolds with the rational homology of defined by Mrowka, Ruberman and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Frøyshov invariant and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct the existence of metrics of positive scalar curvature on certain –manifolds, and to exhibit new classes of homology –spheres of infinite order in the homology cobordism group.
We show that the triply graded Khovanov–Rozansky homology of the torus link stabilizes as . We explicitly compute the stable homology, as a ring, which proves a conjecture of Gorsky, Oblomkov, Rasmussen and Shende. To accomplish this, we construct complexes of Soergel bimodules which categorify the Young symmetrizers corresponding to one-row partitions and show that is a stable limit of Rouquier complexes. A certain derived endomorphism ring of computes the aforementioned stable homology of torus links.
Using the recent theory of noncommutative motives, we compute the additive invariants of orbifolds (equipped with a sheaf of Azumaya algebras) using solely “fixed-point data”. As a consequence, we recover, in a unified and conceptual way, the original results of Vistoli concerning algebraic –theory, of Baranovsky concerning cyclic homology, of the second author and Polishchuk concerning Hochschild homology, and of Baranovsky and Petrov, and Cǎldǎraru and Arinkin (unpublished), concerning twisted Hochschild homology; in the case of topological Hochschild homology and periodic topological cyclic homology, the aforementioned computation is new in the literature. As an application, we verify Grothendieck’s standard conjectures of type and , as well as Voevodsky’s smash-nilpotence conjecture, in the case of “low-dimensional” orbifolds. Finally, we establish a result of independent interest concerning nilpotency in the Grothendieck ring of an orbifold.
This paper opens the study of quasi-isometric embeddings of symmetric spaces. The main focus is on the case of equal and higher rank. In this context some expected rigidity survives, but some surprising examples also exist. In particular there exist quasi-isometric embeddings between spaces and where there is no isometric embedding of into . A key ingredient in our proofs of rigidity results is a direct generalization of the Mostow–Morse lemma in higher rank. Typically this lemma is replaced by the quasiflat theorem, which says that the maximal quasiflat is within bounded distance of a finite union of flats. We improve this by showing that the quasiflat is in fact flat off of a subset of codimension .
We show that the moduli stacks of semistable sheaves on smooth projective varieties are analytic locally on their coarse moduli spaces described in terms of representations of the associated Ext–quivers with convergent relations. When the underlying variety is a Calabi–Yau –fold, our result describes the above moduli stacks as critical loci analytic locally on the coarse moduli spaces. The results in this paper will be applied to the wall-crossing formula of Gopakumar–Vafa invariants defined by Maulik and the author.
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