Related papers: Acylindrical group actions on quasi-trees

Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding…

We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metric…

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are…

Let $A$ be a finite dimensional $Q-$algebra and $\Gamma subset A$ a $Z-$order. We classify those $A$ with the property that $Z^2$ does not embed in $\mathcal{U}(\Gamma)$. We call this last property the hyperbolic property. We apply this in…

Building on work from Sun and Kechris-Quorning, we prove that every acylindrically hyperbolic group $G$ admits a weakly mixing probability measure preserving action $G \curvearrowright (X,\mathcal{B},\mu)$ which is faithful but not…

We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique…

A group $\Gamma$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $\Gamma$ admits a cusp-uniform action on a proper $\delta$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric,…

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure…

We show that if H is a non-elementary hyperbolic commensurated subgroup of infinite index in a hyperbolic group G, then H is virtually a free product of hyperbolic surface groups and free groups. We prove that whenever a one-ended…

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We…

A Cayley graph over a group $G$ is said to be central if its connection set is a normal subset of $G$. We prove that every central Cayley graph over a simple group $G$ has at most two pairwise nonequivalent Cayley representations over $G$…

It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in…

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that its automorphism group ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple…

A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups…

We describe a procedure which verifies that a group given by generators and relators is word-hyperbolic. This procedure always works with a group which is word-hyperbolic, provided there is sufficient memory and time devoted to the problem.…

We show that for any group $G$ that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then $G$ acts properly on a uniformly convex Banach space as well.

In this article, we show that some negative curvature may survive when taking the automorphism group of a finitely generated group. More precisely, we prove that the automorphism group $\mathrm{Aut}(G)$ of a one-ended hyperbolic group $G$…

Let $H$ be a subgroup of a finite non-abelian group $G$ and $g \in G$. Let $Z(H, G) = \{x \in H : xy = yx, \forall y \in G\}$. We introduce the graph $\Delta_{H, G}^g$ whose vertex set is $G \setminus Z(H, G)$ and two distinct vertices $x$…

We prove that for any finitely generated relatively hyperbolic group G and any symmetric endomorphism f of G with relatively quasiconvex image, Fixf is relatively quasiconvex subgroup of G.

We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex…