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We say that a group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the…

Group Theory · Mathematics 2015-05-12 D. Osin

A subgroup of a group $G$ is called algebraic if it can be expressed as a finite union of solution sets to systems of equations. We prove that a non-elementary subgroup $H$ of an acylindrically hyperbolic group $G$ is algebraic if and only…

Group Theory · Mathematics 2017-02-07 Bryan Jacobson

We construct examples of free-by-cyclic hyperbolic groups which fiber in infinitely many ways over Z. The construction involves adding a specialized square 2-cell to a non-positively curved, squared 2-complex defined by labeled oriented…

Group Theory · Mathematics 2014-11-11 TaraLee Mecham , Antara Mukherjee

We prove that a non-elementary relatively hyperbolic group is statistically hyperbolic with respect to every finite generating set. We also establish statistical hyperbolicity for certain direct products of two groups, one of which is…

Group Theory · Mathematics 2016-09-21 Jeremy Osborne , Wen-yuan Yang

We construct and classify all groups, given by triangular presentations associated to the smallest thick generalized quadrangle, that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial…

Group Theory · Mathematics 2019-02-20 Lisa Carbone , Riikka Kangaslampi , Alina Vdovina

Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups…

Group Theory · Mathematics 2015-08-04 Daniel Allcock , Lisa Carbone

By Thurston's hyperbolization theorem, irreducible handlebody-knots are classified into three classes: hyperbolic, toroidal, and atoroidal cylindrical. It is known that a non-trivial handlebody-knot of genus two has a finite symmetry group…

Geometric Topology · Mathematics 2021-04-12 Yi-Sheng Wang

We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is "admitting exponentially many…

Group Theory · Mathematics 2017-10-19 David Hume , Alessandro Sisto

We prove that a hyperbolic group cannot contain a strictly ascending chain of free quasiconvex subgroups of constant rank.

Group Theory · Mathematics 2024-05-24 Jack Kohav , Nir Lazarovich

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a…

Group Theory · Mathematics 2014-02-26 Daniel Groves , Henry Wilton

In this paper, we study the so-called diagram groups. Our main result is that diagram groups are free if and only if they do not contain any subgroup isomorphic to $\mathbb{Z}^2$. As an immediate corollary, we get that hyperbolic diagram…

Group Theory · Mathematics 2015-05-11 Anthony Genevois

In all dimensions $n \ge 5$, we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text{spin}^c$ structure, and in fact we show that there are infinitely many commensurability classes of such manifolds.…

Geometric Topology · Mathematics 2025-03-04 Jacopo G. Chen

We give a combinatorial criterion that implies both the non-strong relative hyperbolicity and the one-endedness of a finitely generated group. We use this to show that many important classes of groups do not admit a strong relatively…

Geometric Topology · Mathematics 2007-05-23 James W. Anderson , Javier Aramayona , Kenneth J. Shackleton

We prove that relatively hyperbolic groups do not have Lafforgue strong Property $(T)$ with respect to Hilbert spaces. To do so we construct an unbounded affine representation of such groups, whose linear part is of polynomial growth of…

Group Theory · Mathematics 2023-12-25 Hermès Lajoinie-Dodel

We construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a construction of groups which are of type $F_n$ but…

Group Theory · Mathematics 2018-02-07 Robert Kropholler

We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG…

Group Theory · Mathematics 2012-09-21 James Howie , Gerald Williams

We produce an example demonstrating that every finitely generated relatively hyperbolic group with respect to a collection of Hopfian subgroups need not be Hopfian. This answers a question of Osin \cite[Problem 5.5]{Osin} in the negative.

Group Theory · Mathematics 2023-04-21 Jan Kim , Donghi Lee

Italiano-Martelli-Migliorini recently constructed hyperbolic groups which have non-hyperbolic subgroups of finite type. Using a closely related construction, Llosa Isenrich-Martelli-Py constructed hyperbolic groups with subgroups of type…

Group Theory · Mathematics 2022-08-31 Daniel Groves , Jason Fox Manning

We prove the following boundary-theoretic characterization of relatively hyperbolic groups. Let $G$ be a finitely generated group with a finite collection $\mathcal{H}$ of finitely generated subgroups, and let $G^h$ denote the associated…

Geometric Topology · Mathematics 2026-03-25 Vyshnav PT , Pranab Sardar , Rana Sardar

We construct nonlinear hyperbolic groups which are large, torsion-free, one-ended, and admit a finite $K(\pi,1)$. Our examples are built from superrigid cocompact rank one lattices via amalgamated free products and HNN extensions.

Group Theory · Mathematics 2019-04-24 Richard Canary , Matthew Stover , Konstantinos Tsouvalas