Related papers: CAT(0) cube complexes and asymptotically rigid map…

We show that for $n \geq 3$, there are torsion-free CAT(0) groups with visual boundaries that embed into $S^n$ but which are not virtually the fundamental group of a compact aspherical $n+1$-manifold. The groups are the CAT(0) and not…

In this paper, we study boundary actions of CAT(0) spaces from a point of view of topological dynamics and $C^*$-algebras. First, we investigate the actions of right-angled Coexter groups and right-angled Artin groups with finite defining…

We initiate systematic study of EZ-structures (and associated boundaries) of groups acting on spaces that admit consistent and conical (equivalently, consistent and convex) geodesic bicombings. Such spaces recently drew a lot of attention…

We consider geodesically convex optimization problems involving distances to a finite set of points $A$ in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the…

We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichm\"uller spaces are stably approximated by a CAT(0) cube complexes, strengthening a result of Behrstock-Hagen-Sisto. As applications, we prove…

Following the research from the paper "Triangles, squares and geodesics" (arXiv:0910.5688) of Rena Levitt and Jon McCammond we investigate the properties of groups acting on CAT(0) triangle-square complexes, focusing mostly on…

We prove a strengthened sector lemma for irreducible, finite-dimensional, locally finite, essential, cocompact CAT(0) cube complexes under the additional hypothesis that the complex is \emph{hyperplane-essential}; we prove that every…

We prove that the class of reflexive asymptotic-$c_0$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_0$ space $Y$, then $X$ is also reflexive and asymptotic-$c_0$. In order…

Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type…

Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper…

We present examples of prequantizations over integral symplectic manifolds which admit infinitely many smoothly trivial contact mapping classes. These classes are given by the connected components of the strict contactomorphism group which…

This paper provides uniform bounds on the asymptotic regularity for iterations associated to a finite family of nonexpansive mappings. We obtain our quantitative results in the setting of $(r,\delta)$-convex spaces, a class of geodesic…

For each d we construct CAT(0) cube complexes on which Cremona groups rank d act by isometries. From these actions we deduce new and old group theoretical and dynamical results about Cremona groups. In particular, we study the dynamical…

We compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group. The surface is compact, oriented with one boundary component. This…

We show that the affine cactus group is a CAT$(0)$ group for all degrees. Furthermore, we show that the affine cactus group $AJ_3$ of degree three is a hyperbolic group.

In [17], we introduced ``picture groups'' and computed the cohomology of the picture group of type $A_n$. This is the same group what was introduced by Loday [20] where he called it the ``Stasheff group''. In this paper, we give an…

This paper completely determines the non-amenability of the mapping class groups of infinite-type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non-amenable stabiliser of a point…

We consider the action of the group $\mathrm{PGL}_4(K)$ on the smooth cubic surfaces of $\mathbb{P}^3_K$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with non…

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first…

We study the ergodic properties for a class of quantized toral automorphisms, namely the cat and Kronecker maps. The present work uses and extends the results of [KL]. We show that quantized cat maps are strongly mixing, while Kronecker…