We characterize hyperbolic groups in terms of quasigeodesics in the Cayley graph forming regular languages. We also obtain a quantitative characterization of hyperbolicity of geodesic metric spaces by the non-existence of certain local $(3,0)$-quasigeodesic loops. As an application, we make progress towards a question of Shapiro regarding groups admitting a uniquely geodesic Cayley graph.

We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.

We construct examples of quasi-isometric embeddings of word hyperbolic groups into $\mathsf {SL}(d,\mathbb {R})$ for $d \geq 4$ which are not limits of Anosov representations into $\mathsf {SL}(d,\mathbb {R})$. As a consequence, we conclude that an analogue of the density theorem for $\mathsf {PSL}(2,\mathbb {C})$ does not hold for $\mathsf {SL}(d,\mathbb {R})$ when $d \geq 4$.

We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$, the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$, which implies hyperbolicity.

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.

In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups and the lamplighter group.

Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly bi-Lipschitz Equivalences (SBE) are a weak variant of quasi-isometries, with the only requirement of still inducing bi-Lipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of boundary extensions of SBEs, reminiscent of quasi-Möbius (or quasisymmetric) mappings. We give a dimensional invariant of the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answering a question of Druţu.

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected 2-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of attachingmaps of 2-cells, and finitely many 2-cells adjacent to any edge must have a fine 1-skeleton. We provide a positive answer to this question. We revisit a homological characterization of relative hyperbolicity and show that a group $G$ is hyperbolic relative to a collection of subgroups $P$ if and only if $G$ acts cocompactly with finite edge stabilizers on a connected 2-dimensional cell complex with a linear homological isoperimetric inequality and $P$ is a collection of representatives of conjugacy classes of vertex stabilizers.

There exist infinite finitely presented torsion-free groups G such that Aut(G) and Out(G) are torsion free but G has an automorphism sending some non-trivial element to its inverse.

Given a group automorphism $\phi :\,\Gamma \,\to \,\Gamma $ , one has an action of $\Gamma $ on itself by $\phi $ -twisted conjugacy, namely, $g.x\,=\,gx\phi ({{g}^{-1}})$ . The orbits of this action are called $\phi $ -twisted conjugacy classes. One says that $\Gamma $ has the ${{R}_{\infty }}$ -property if there are infinitely many $\phi $ -twisted conjugacy classes for every automorphism $\phi $ of $\Gamma $ . In this paper we show that $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ and its congruence subgroups have the ${{R}_{\infty }}$ -property. Further we show that any (countable) abelian extension of $\Gamma $ has the ${{R}_{\infty }}$ -property where $\Gamma $ is a torsion free non-elementary hyperbolic group, or $\text{SL(}n\text{,}\mathbb{Z}\text{)},\text{Sp(2}n\text{,}\mathbb{Z}\text{)}$ or a principal congruence subgroup of $\text{SL(}n\text{,}\mathbb{Z}\text{)}$ or the fundamental group of a complete Riemannian manifold of constant negative curvature.

We show that every non-elementary hyperbolic group $\G $ admits a proper affine isometric action on $L^p(\bd \G \times \bd \G )$, where $\bd \G $ denotes the boundary of $\G $ and $p$ is large enough. Our construction involves a $\G $-invariant measure on $\bd \G \times \bd \G $ analogous to the Bowen–Margulis measure from the ${\rm CAT}(-1)$ setting, as well as a geometric, Busemann-type cocycle. We also deduce that $\G $ admits a proper affine isometric action on the first $\ell ^p$-cohomology group $H^1_{(p)}(\G )$ for large enough $p$.

Nous construisons une $KK$-théorie pour les algèbres de Banach équivariante par l’action d’un groupoïde et nous montrons la conjecture de Baum–Connes à coefficients commutatifs pour les groupes hyperboliques et pour les groupoïdes de Poincaré des feuilletages à base compacte qui peuvent être munis d’une métrique riemannienne longitudinale à courbure sectionnelle strictement négative.

We construct a $KK$-theory for Banach algebras, equivariant with respect to the action of a groupoid. We prove the Baum–Connes conjecture with commutative coefficients for hyperbolic groups and for the Poincaré groupoids of foliations with a compact base and a longitudinal Riemannian metrics with negative sectional curvature.