In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering [6, Question D] posed by Behrstock

Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one universal cover contains a nonplanar graph, while the visual boundary of the other does not. In contrast, we prove for any two locally CAT(0) metrics on the Croke–Kleiner

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves’ lengths do we really need to know? It is a result of Duchin, Leininger, and Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We generalize the notion of simple curves to that of $q$-simple curves, for any positive

Let $A$ be a finite set and $G$ a group. A closed subset $X$ of $A^G$ is called a subshift if the action of $G$ on $A^G$ preserves $X$. If $K$ is a closed subset of $A^G$ such that membership in $K$ is determined by looking at a fixed finite set of coordinates, and $X$ is the intersection of all translates of $K$ under the action of $G$, then $X$ is called a subshift of finite type (SFT). If an SFT

We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag–Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of $F_2$ and nonsurjective fully irreducible endomorphisms of $F_n$. We also give a framework for extending the theorem to all injective

We consider a strictly developable simple complex of finite groups $G(\mathcal Q)$. We show that Bestvina's construction for Coxeter groups applies in this more general setting to produce a complex that is equivariantly homotopy equivalent to the standard development. When $G(\mathcal Q)$ is non-positively curved, this implies that the Bestvina complex is a cocompact classifying space for proper actions

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\binom{n+1}{3}$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.

Let $\Sigma$ be a compact orientable surface of finite type with at least one boundary component. Let $\Gamma \leq \mathrm{Mod}(\Sigma)$ be a non virtually solvable subgroup. We answer a question of Lubotzky by showing that there exists a finite dimensional homological representation $\rho$ of $\mathrm{Mod}(\Sigma)$ such that $\rho(\Gamma)$ is not virtually solvable. We then apply results of Lubotzky

For a one-ended right-angled Artin group, we give an explicit description of its JSJ tree of cylinders over infinite cyclic subgroups in terms of its defining graph. This is then used to classify certain right-angled Artin groups up to quasi-isometry. In particular, we show that if two right-angled Artin groups are quasi-isometric, then their JSJ trees of cylinders are weakly equivalent. Although the

Applying the techniques developed in [1], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov–Lawson–Thurston conjecture [6], such bundles $M \to S$ should satisfy the inequality $|eM/\chi S|\leq 1$, where $eM$ stands for the Euler number of the bundle and $\chi S$, for the Euler characteristic of the surface

We show that for many semi-hyperbolic groups the decomposition into conjugacy classes of the Hochschild homology of the $\ell^1$-rapid decay group algebra is injective.

We complete the classification of hyperelliptic threefolds, describing in an elementary way the hyperelliptic threefolds with group $D_4$. These are algebraic and form an irreducible 2-dimensional family.

We are interested in the Guivarc’h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality is an equality, then the Green distance is roughly similar to the word distance, generalizing results of Blachère, Haïssinsky, and Mathieu for hyperbolic groups

We prove that a relatively hyperbolic pair $(G,P)$ has Bowditch boundary a 2-sphere if and only if it is a 3-dimensional Poincare duality pair. We prove this by studying the relationship between the Bowditch and Dahmani boundaries of relatively hyperbolic groups.

In this note we refine examples by Aka from arithmetic to $S$-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \geq 2$.

The recent breakthrough works [9, 11, 12] which established the amenability for new classes of groups, lead to the following question: is the action $W(\mathbb Z^d) \curvearrowright \mathbb Z^d$ extensively amenable? (Where $W(\mathbb Z^d)$ is the wobbling group of permutations $\sigma\colon \mathbb Z^d \to \mathbb Z^d$ with bounded range). This is equivalent to asking whether the action $(\mathbb

We show that integral foliated simplicial volume of closed manifolds gives an upper bound for the cost of the corresponding fundamental groups.

We classify pro-$p$ Poincaré duality pairs in dimension two. We then use this classification to build a pro-$p$ analogue of the curve complex and establish its basic properties. We conclude with some statements concerning separability properties of the mapping class group.

In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $B\subseteq A$ of $C^*$-algebras is isomorphic to the canonical inclusion of $\ell^\infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for “Roe Cartans” inside uniform Roe algebras up

This work is a natural follow-up of the article [5]. Given a group-word $w$ and a group $G$, the verbal subgroup $w(G)$ is the one generated by all $w$-values in $G$. The word $w$ is called concise if $w(G)$ is finite whenever the set of $w$-values in $G$ is finite. It is an open question whether every word is concise in residually finite groups. Let $w=w(x_1,\ldots,x_k)$ be a multilinear commutator

The space of deformations of the integer Heisenberg group under the action of Aut$(\mathbf H(\mathbb R))$ is a homogeneous space for a non-reductive group. We analyze its structure as a measurable dynamical system and obtain mean and variance estimates for Heisenberg lattice point counting in measurable subsets of $\mathbb R^3$; in particular, we obtain a random Minkowski-type theorem. Unlike the Euclidean

A group homomorphism $i\colon H \to G$ is a localization of $H$, if for every homomorphism $\varphi\colon H\to G$ there exists a unique endomorphism $\psi\colon G\to G$ such that $i \psi=\varphi$ (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable

We investigate polynomial endomorphisms of graph $C^*$-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the

Every pseudo-Anosov mapping class $\phi$ defines an associated veering triangulation $\tau_\phi$ of a punctured mapping torus. We show that generically, $\tau_\phi$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination

We collect three observations on the homology for Smale spaces defined by Putnam. The definition of such homology groups involves four complexes. It is shown here that a simple convergence theorem for spectral sequences can be used to prove that all complexes yield the same homology. Furthermore, we introduce a simplicial framework by which the various complexes can be understood as suitable "symmetric"

The congruence subgroup problem for a finitely generated group $\Gamma$ asks whether the map $\widehat{\mathrm{Aut}(\Gamma)}\to \mathrm{Aut}(\widehat{\Gamma})$ is injective, or more generally, what is its kernel $C(\Gamma)$? Here $\widehat{X}$ denotes the profinite completion of $X$. It is well known that for finitely generated free abelian groups $C(\mathbb{Z}^{n})=\{ 1\}$ for every $n\geq3$, but

In his work on the Farrell–Jones Conjecture, Arthur Bartels introduced the concept of a "finitely $\mathcal F$-amenable" group action, where $\mathcal F$ is a family of subgroups. We show how a finitely $\mathcal F$-amenable action of a countable group $G$ on a compact metric space, where the asymptotic dimensions of the elements of $\mathcal F$ are bounded from above, gives an upper bound for the

We construct a one parameter family of ICC groups $\{G_t\}_{t > 1}$, with the property that the group factor $L(G_t)$ is isomorphic to the interpolated free group factor $L(\mathbb F_t):=L(\mathbb{F}_2)^{1/\sqrt{t-1}}$, for all $t$. Moreover, the groups $G_t$ have fixed cost $t$, are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost.

For a non-elementary discrete isometry group $G$ of divergence type acting on a proper geodesic $delta$-hyperbolic space, we prove that its Patterson measure is quasi-invariant under the normalizer of $G$. As applications of this result, we have: (1) under a minor assumption, such a discrete group $G$ admits no proper conjugation, that is, if the conjugate of $G$ is contained in $G$, then it coincides

Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1 \in S = S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x \in X$, where $G = \langle S \rangle$. We show in this setting that a number of conditions are equivalent: (a) $G$ acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every

Given a countable residually finite group $\Gamma$, we write $\Gamma_n \to e$ if $(\Gamma_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $\Gamma_n$'s contains only the unit element $e$ of $\Gamma$. Given a $\Gamma$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} \label{eq:1a} \chi (\Gamma_n , M) = \prod_i |H_i (\Gamma_n

We address the problem of which functions can arise as Dehn functions of Kähler groups. We explain why there are examples of Kähler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a Kähler group which has Dehn function bounded below by a cubic function and above by $n^6$. As a consequence we obtain that for a compact Kähler manifold

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus $g \geq 3$ without punctures is Sp$_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz’s results on $\mathbb C$-linear representations of mapping class groups to projective representations over any field.

We extend the notion of Universal Minimal Flows to groupoid actions of locally trivial groupoids. We also prove that any $G$-bundle with compact fibers has a global section if $G$ is extremely amenable.

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when (1) the step distribution has exponential tail and (2) the action on $X$ is acylindrical

We realize lamplighter groups $A\wr \mathbb Z$, with $A$ a finite abelian group, as automaton groups via affine transformations of power series rings with coefficients in a finite commutative ring. Our methods can realize $A\wr \mathbb Z$ as a bireversible automaton group if and only if the 2-Sylow subgroup of $A$ has no multiplicity one summands in its expression as a direct sum of cyclic groups of

By Namazi and Johnson’s results, for any distance at least 4 Heegaard splitting, its mapping class group is finite. In contrast, Namazi showed that for a weakly reducible Heegaard splitting, its mapping class group is infinite; Long constructed an irreducible Heegaard splitting where its mapping class group contains a pseudo anosov map. Thus it is interesting to know that for a strongly irreducible

We study the large-scale geometry of Weil–Petersson space, that is, Teichmüller space equipped with theWeil–Petersson metric. We show that this admits a natural coarse median structure of a specific rank. Given that this is equal to the maximal dimension of a quasi-isometrically embedded euclidean space,we recover a result of Eskin,Masur and Rafi which gives the coarse rank of the space. We go on

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We then use this result to show in each dimension $d\geqslant 3$ there are examples

If $G$ is a Grigorchuk–Gupta–Sidki group defined over a $p$-adic tree, where $p$ is an odd prime, we study the existence of Beauville surfaces associated to the quotients of $G$ by its level stabilizers $\mathrm {st}_G(n)$. We prove that if $G$ is periodic then the quotients $G/\mathrm {st}_G(n)$ are Beauville groups for every $n\geq 2$ if $p\geq 5$ and $n\geq 3$ if $p = 3$. In this case, we further

We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.

We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ by an arbitrary subgroup $G$ of $F_n$. Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on $G \backslash {\rm Cay}(F_n)$ to the

In this paper, we address the following question: when is a finite $p$-group $G$ self-similar, i.e. when can $G$ be faithfully represented as a self-similar group of automorphisms of the $p$-adic tree? We show that, if $G$ is a self-similar finite $p$-group of rank $r$, then its order is bounded by a function of $p$ and $r$. This applies in particular to finite $p$-groups of a given coclass. In the