Let $G$ be a torsion-free hyperbolic group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. We want to understand the possible growth types for the word length of $f^n(g)$, where $g$ is an element of $G$. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We address the general case of a torsion-free hyperbolic group $G$; we show that every element in $G$ has a well-defined exponential growth rate under iteration of $f$, and that only finitely many exponential growth rates arise as $g$ varies in $G$. In addition, we show the following dichotomy: every element of $G$ grows either exponentially fast or polynomially fast under iteration of $f$.
This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.[-]

Bestvina-Mess showed that the duality properties of a group $G$ are encoded in any boundary that gives a Z-compactification of $G$. For example, a hyperbolic group with Gromov boundary an $n$-sphere is a PD$(n+1)$ group. For relatively hyperbolic pairs $(G,P)$, the natural boundary - the Bowditch boundary - does not give a Z-compactification of G. Nevertheless we show that if the Bowditch boundary of $(G,P)$ is a 2-sphere, then $(G,P)$ is a PD(3) pair.
This is joint work with Genevieve Walsh.[-]

We are interested in the structure of the set of homomorphisms from a fixed (but arbitrary) finitely generated group G to the groups in some fixed family (such as the family of 3-manifold groups). I will explain what one might hope to say in different situations, and explain some applications to relatively hyperbolic groups and acylindrically hyperbolic groups, and some hoped-for applications to 3-manifold groups.
This is joint work with Michael Hull and joint work in preparation with Michael Hull and Hao Liang.[-]

Let $G$ be a hyperbolic group. Its boundary is a topological invariant within the quasi-isometry class of $G$ but it is far from being a complete invariant, e.g. a random group at density ¡1/2 is hyperbolic (Gromov) and its boundary is homeomorphic to the Menger curve (Dahmani-Guirardel-Przytycki) but Mackay proved that there are infinitely many quasi-isometry classes of random groups at density d for small enough d.
We discuss the conformal dimension of a hyperbolic group, a quasi-isometry invariant introduced by Pansu. Paulin proved that this is a complete $QI$ invariant of the group. We discuss a technique of Pansu and Bourdon for bounding the conformal dimension from below. We then relate this technique to the family of hyperbolic free by cyclic groups. This is work in progress towards the ultimate goal of showing that there are infinitely many $QI$ classes of free by cyclic groups.
This is joint work with Bestvina, Hilion and Stark[-]

(joint work with Michael Handel) $Out(F_{n}) := Aut(F_{n})/Inn(F_{n})$ denotes the outer automorphism group of the rank n free group $F_{n}$. An element $f$ of $Out(F_{n})$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_{n}), n > 2$, of elements with non-linear polynomial growth is a feature of $Out(F_{n})$ not shared by mapping class groups of surfaces.
To avoid some finite order behavior, we restrict attention to the subset $UPG(F_{n})$ of $Out(F_{n})$ consisting of polynomially growing elements whose action on $H_{1}(F_{n}, Z)$ is unipotent. In particular, if $f$ is polynomially growing and acts trivially on $H_{1}(F_{n}, Z_{3})$ then $f $ is in $UPG(F_{n})$ and further every polynomially growing element of $Out(F_{n})$ has a power that is in $UPG(F_{n})$. The goal of the talk is to describe an algorithm to decide given $f,g$ in $UPG(F_{n})$ whether or not there is h in $Out(F_{n})$ such that $hf h^{-1} = g$.
The conjugacy problem for linearly growing elements of $UPG(F_{n})$ was solved by Cohen-Lustig. Krstic-Lustig-Vogtmann solved the case of linearly growing elements of $Out(F_{n})$.
A key technique is our use of train track representatives for elements of $Out(F_{n})$, a method pioneered by Bestvina-Handel in the early 1990s that has since been ubiquitous in the study of $Out(F_{n})$.[-]

A well-known result of Davis-Januszkiewicz is that every right-angled Artin group (RAAG) is commensurable to some rightangled Coxeter group (RACG). In this talk we consider the converse question: which RACGs are commensurable to some RAAG? To do so, we investigate some natural candidate RAAG subgroups of RACGs and characterize when such subgroups are indeed RAAGs. As an application, we show that a 2-dimensional, one-ended RACG with planar defining graph is quasiisometric to a RAAG if and only if it is commensurable to a RAAG. This talk is based on work joint with Pallavi Dani.[-]

If a group G acts isometrically on a metric space X and Y is any metric space that is quasi-isometric to X, then G quasi-acts on Y. A fundamental problem in geometric group theory is to straighten (or quasi-conjugate) a quasi-action to an isometric action on a nice space. We will introduce and investigate discretisable spaces, those for which every cobounded quasi-action can be quasi-conjugated to an isometric action of a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups have this property, but it holds much more generally. For instance, we show that every hyperbolic group is either commensurable to a cocompact lattice in rank one Lie group, or it is discretisable.
We give several applications and indicate possible future directions of this ongoing work, particularly in showing that normal and almost normal subgroups are often preserved by quasi-isometries. For instance, we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is Z-by-hyperbolic. We also show that within the class of residually finite groups, the class of central extensions of finitely generated abelian groups by hyperbolic groups is closed under quasi-isometries.[-]