arXiv research

Generalizes Nielsen equivalence theorem to hyperbolic group extensions.

Hyperbolicity proven for a specific type of group extension.

New groups are hyperbolic and rigid in mapping class groups.

Outer automorphism group of hyperbolic groups is HHG under certain conditions.

Extends growth properties of hyperbolic groups to their extensions.

Multicurve stabilizers' extensions are hierarchically hyperbolic.

Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.

New lattice extensions of Schottky groups in hyperbolic space.

New groups with unique properties discovered.

In this note, we prove that a random extension of either the free group $F_N$ of rank $N\ge3$ or of the fundamental group of a closed, orientable surface $S_g$ of genus $g\ge2$ is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either Out$(F_N)$ or Mod$(S_g)$ generated by $k$ independ…

New hyperbolic groups from ping-pong automorphisms.

We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\ma…

Extensions of Veech groups using hierarchical hyperbolic spaces.

The paper studies a group action on a hyperbolic space derived from a lattice Veech group.

Study the geometry of graph product extension graphs.

Survey connects hyperbolic groups to manifolds and Kleinian groups.

The paper studies groups with proper actions on finite products of hyperbolic spaces.

Given a finitely generated subgroup $Γ\le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension $1 \to \mathbb{F} \to E_Γ \to Γ\to 1$. We give sufficient conditions for when the extension $E_Γ$ is hyperbolic. In particular,…

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theo…

The paper defines conditions for a free-by-free group to be hyperbolic.

Linear progress observed in fibered 3-manifold embeddings.

Proves Farrell--Jones Conjecture for hyperbolic groups and their automorphisms.

We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain …

This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let $F_N$ denote a free groups of finite rank $N\ge 3$ and consider a \emph{convex cocompact} subgroup $Γ\le Out(F_N)$, i.e. one for which the orbit map from $Γ$ into the free factor comp…

Logarithmic separation profile in hyperbolic groups shows hierarchical structure.

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex coc…

We study Nielsen equivalence classes of generating pairs of Kleinian groups and HNN-extensions. We establish the following facts: - Hyperbolic 2-bridge knot groups have infinitely many Nielsen classes of generating pairs. - For any natural number N there is a closed hyperbolic 3-manifold whose fundamental group has N d…

New conditions ensure surface group extensions are non-positively curved.

We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Com…

New group not biautomatic, geometrically constructed.

Extends group actions on metric spaces, preserving properties.

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infi…

Surjectivity of Cannon-Thurston map proven for metric graph bundles.

Extends Anosov subgroup definitions to more general groups.

We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R-tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperboli…

Extends confining subset theory to describe hyperbolic actions of solvable groups with higher rank abelianizations.

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $π_1(M)$.…

Study on finiteness property of right-angled Artin groups actions on extension graphs.

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron-Frobenius theory. We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions us…

This paper is a more succinct version of the author's 1993 UCLA mathematics thesis. It proves that any group quasi-isometric to the product of the hyperbolic plane with the real line is a finite extension of a cocompact lattice in either the isometry group of the product of the hyperbolic plane with the real line or th…

Stable actions of hyperbolic groups on their boundaries.

New groups with Menger curve boundaries found.

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most $9$-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends resul…

We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface…

The paper constructs Anosov representations for specific types of groups.

Proves one-relator groups with negative immersions are hyperbolic and virtually special.

We present an extension of Dunwoody's theory of tracks and use it to prove an analogue of the annulus theorem for hyperbolic groups.