We give another proof of a theorem of Hatcher and Vogtmann stating that the sequence $Aut(F_n)$ satisfies integral homological stability. The paper is for the most part expository, and we also explain Quillen's method for proving homological stability.

Suppose we have a one-ended finitely presented group with a purely loxodromic action on a Gromov hyperbolic space satisfying an acylindricity condition. We show that, given a finite generating set, there is an automorphism of the group, and some point in the space which is moved a bounded distance by each of the images of the generators under the automorphism. Here the bound depends only on the group, generating set, and constants of hyperbolicity and acylindricity. With results from elsewhere, this implies that, up to conjugacy, there can only be finitely many purely pseudoanosov subgroups of a mapping class group that are isomorphic to a given one-ended finitely presented group.

We address the question of whether the property of being virtually special (in the sense of Haglund and Wise) is algorithmically decidable for finite, non-positively curved cube complexes. Our main theorem shows that it cannot be decided by examining one hyperplane at a time. Specifically, we prove that there does not exist an algorithm that, given a compact non-positively curved squared 2-complex $X$ and a hyperplane $H$ in $X$, will decide whether or not there is a finite-sheeted cover of $X$ in which no lift of $H$ self-osculates.

We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lipschitz constant decays exponentially in the distance from the cone-singularity. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves.

Every closed subset of the Riemann sphere can be approximated in the Hausdorff topology by the Julia set of a rational map.

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathsf{SL}(n+1, \mathbb{R})$ or $\mathsf{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds for $n \geq 2$. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study the natural conditions on eigenvalues of holonomy representations of ends when these ends are manageably understandable. We will show that only the radial or totally geodesic ends of lens shape or horospherical ends exist for strongly irreducible properly convex real projective orbifolds under some suitable conditions. The purpose of this article is to announce these results.

In a previous work, we remarked that the conjugacy problem for pairs of atoroidal automorphisms of a free group was solvable by mean of the isomorphism problem for hyperbolic groups and an orbit problem for the automorphism group of their suspensions (i.e. their semidirect product with $\mathbb{Z}$ for the relevant structural automorphism).

We consider the same problem a few more automorphisms of free groups, those that produce relatively hyperbolic suspensions that do not split over a parabolic subgroup.

We characterize when (and how) a Right-Angled Artin group splits nontrivially over an abelian subgroup.

Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property.

We survey embeddability results related to RAAGs (right-angled Artin groups) and various automorphism groups of manifolds. We give two different methods of embedding a RAAG to into another, and deduce that every RAAG embeds into some braid groups. This gives the unsolvability of the isomorphism problem for finitely presented subgroups of braid groups. Also, we prove that every RAAG is a quasi-isometrically embedded subgroup of the symplectomorphism groups of the disk and the sphere, given with suitable $L^p$ metrics. Finally, we embed RAAGs in the smooth diffeomorphism group of the real line. These results reveal many closed hyperbolic manifold subgroups of diffeomorphism groups of manifolds.

We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise. Our goal is to go through the works of some of the eminent mathematicians from the past and to include them in a modern perspective. Putting together results in the three constant-curvature geometries and highlighting the analogies between them is mathematically as well as aesthetically very appealing.

The aim of this note is to give the simplest possible proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, and more specifically that their curve graphs are hyperbolic and that pseudo-Anosovs act on them as loxodromic WPDs.

This note contains a newly streamlined version of the original proof that Outer space is contractible.