Related papers: Fibonacci groups, F(2,n), are hyperbolic for n odd…

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of…

We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular $\mathrm{Aut}(\mathbb{F}_n)$ is acylindrically hyperbolic for every $n\ge 2$. More generally, if $G$ is a…

The Fibonacci groups $F(n)$ are known to exhibit significantly different behaviour depending on the parity of $n$. We extend known results for $F(n)$ for odd $n$ to the family of Fractional Fibonacci groups $F^{k/l}(n)$. We show that for…

We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n) in {(7 + 5k,5), (8 + 5k,5)} where k is greater than or equal to 0. This together with previously known results yields a complete classification of the finite F(r,n),…

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n-1,1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined…

We show that properties $F_n$ and $FP_n$ hold for a relatively hyperbolic group if and only if they hold for all the peripheral subgroups. As an application we show that there are at least countably many distinct quasi-isometry classes of…

In this article, we prove that if a finitely presented group has an asymptotic cone which is tree-graded with respect to a precise set of pieces then it is relatively hyperbolic. This answers a question of M. Sapir.

The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n > 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n =…

Let $G$ be a finite group of odd order, $\F$ a finite field of odd characteristic $p$ and $\B$ a finite--dimensional symplectic $\F G$-module. We show that $\B$ is $\F G$-hyperbolic, i.e., it contains a self--perpendicular $\F G$-submodule,…

Let G be a graph of hyperbolic groups with 2-ended edge groups. We show that G is hierarchically hyperbolic if and only if G has no distorted infinite cyclic subgroup. More precisely, we show that G is hierarchically hyperbolic if and only…

Let $G$ be a finitely presented group, and let $H$ be a subgroup of $G$. We prove that if $H$ is acylindrically hyperbolic and existentially closed in $G$, then $G$ is acylindrically hyperbolic. As a corollary, any finitely presented group…

Let $Sp(2,1)$ be the isometry group of the quaternionic hyperbolic plane ${{\bf H}_{\mathbb H}}^2$. An element $g$ in $Sp(2,1)$ is `hyperbolic' if it fixes exactly two points on the boundary of ${{\bf H}_{\mathbb H}}^2$. We classify pairs…

We prove that for a finitely generated subgroup $H$ of a word-hyperbolic group $G$ the Frattini subgroup $F(H)$ of $H$ is finite.

We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every…

We call a log variety (X, D) algebraically hyperbolic if there exists a positive number e such that 2g(C) - 2 + i(C, D) >= e deg(C) for all curves C on X, where i(C, D) is the number of the intersections between D and the normalization of…

We prove that for any finitely generated relatively hyperbolic group G and any symmetric endomorphism f of G with relatively quasiconvex image, Fixf is relatively quasiconvex subgroup of G.

Bestvina-Feighn-Handel show that for finitely many generic and independent hyperbolic automorphisms $\phi_1, \cdots, \phi_r$ of $F_n$, the resulting extension $F_n \rtimes F_r$ is hyperbolic. This paper generalizes the above statement to…

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We…

We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups…

We prove that an automorphism $\phi:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.