Related papers: Hierarchically hyperbolic groups and uniform expon…

We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse…

We study the set of homomorphisms from a fixed finitely generated group into a family of groups which are `uniformly acylindrically hyperbolic'. Our main results reduce this study to sets of homomorphisms which do not diverge in an…

A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions.…

We prove that the outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group (HHG), under mild orientability conditions on the associated JSJ decomposition. This is done by proving that a…

Given a finitely generated subgroup $\Gamma \le \mathrm{Out}(\mathbb{F})$ 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_{\Gamma} \to…

We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily…

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific…

We study the homomorphisms from a fixed finitely generated group to strictly acylindrical colorable hierarchically hyperbolic groups. We prove that any such group is equationally noetherian.

In this paper, we establish that the mapping torus of a one-ended torsion-free hyperbolic group exhibits a quadratic isoperimetric inequality.

We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric…

Given a finite rank free group $\mathbb{F}$ of $\mathsf{rank}(\mathbb{F})\geq 3$, we show that the mapping torus of $\phi$ is (strongly) relatively hyperbolic if $\phi$ is exponentially growing. We combine our result with the work of…

A classical result of J\o rgensen and Thurston shows that the set of volumes of finite volume complete hyperbolic $3$-manifolds is a well-ordered subset of the real numbers of order type~$\omega^\omega$; moreover, each volume can only be…

We study the asymptotic behavior of the homotopy groups of simply connected finite $p$-local complexes, and define a space to be locally hyperbolic if its homotopy groups have exponential growth. Under some certain conditions related to the…

We study the growth rate of a sequence which measures the uniform norm of the differential under the iterates of maps. On symplectically hyperbolic manifolds, we show that this sequence has at least linear growth for every non-identical…

We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors. Inspired by the classical theory of…

Behrstock, Hagen, and Sisto classified 3-manifold groups admitting a hierarchically hyperbolic space structure. However, these structures were not always equivariant with respect to the group. In this paper, we classify 3-manifold groups…

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…

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index…

We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan's property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those…

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively…