相关论文: A simple criterion for non-relative hyperbolicity …

Let $G$ be an abelian group. We prove that a group $G$ admits a Hausdorff group topology $\tau$ such that the von Neumann radical $\mathbf{n}(G, \tau)$ of $(G, \tau)$ is non-trivial and finite iff $G$ has a non-trivial finite subgroup. If…

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…

For several instances of metric largeness like enlargeability or having hyperspherical universal covers, we construct non-large vector subspaces in the rational homology of finitely generated groups. The functorial properties of this…

We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely…

Motivated by the notion of boundary for hyperbolic and $CAT(0)$ groups, M. Bestvina in "Local Homology Properties of Boundaries of Groups" introduced the notion of a (weak) $\mathcal Z$-structure and (weak) $\mathcal Z$-boundary for a group…

We study the boundaries of relatively hyperbolic HHGs. Using the simplicial structure on the hierarchically hyperbolic boundary, we characterize both relative hyperbolicity and being thick of order 1 among HHGs. In the case of relatively…

We show that first-order formulae are concise in acylindrically hyperbolic groups and certain extensions thereof. We study further classes of groups, including Burnside groups, icc groups, groups with the `Big Powers' condition, torus knot…

In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at…

We give conditions of an extension of a free group to be hyperbolic and relatively hyperbolic using the dynamics of the action of $\out$ on the complex of free factors combined with the weak attraction theory. We work with subgroups of…

We construct hyperbolic groups with the following properties: The boundary of the group has big dimension, it is separated by a Cantor set and the group does not split. This shows that Bowditch's theorem that characterizes splittings of…

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…

Assuming that every hyperbolic group is residually finite, we prove the congruence subgroup property for mapping class groups of hyperbolic surfaces of finite type. Under the same assumption, it follows that profinitely equivalent…

If there is a non-residually finite hyperbolic group, then there is a non-residually finite rigid hyperbolic group.

Our first result gives a partial converse to a well-known theorem of A. Ancona for hyperbolic groups. We prove that a group $G$, equipped with a symmetric probability measure whose finite support generates $G$, is hyperbolic if it is…

This paper allows one to obtain a criterion for the existence of a projectively invariant measure formulated in terms of combinatorial properties of a group (amenability of some canonical quotient group). Such necessary and sufficient…

We define a family of groups that generalises Thompson's groups $T$ and $G$ and also those of Higman, Stein and Brin. For groups in this family we descrine centralisers of finite subgroups and show, that for a given finite subgroup $Q$,…

We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete.…

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective…

A class of one-relator groups such that every group in the class is determined by a triple of integers and is an HNN-extension of some Baumslag -- Solitar group is considered. A criterion for two groups in this class to be isomorphic and…

For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not…