Related papers: Top dimensional quasiflats in $CAT(0)$ cube comple…

The class of quasi-median graphs is a generalisation of median graphs, or equivalently of CAT(0) cube complexes. The purpose of this thesis is to introduce these graphs in geometric group theory. In the first part of our work, we extend the…

As a consequence of Kirchberg's work, Connes' Embedding Conjecture is equivalent to the property that every homomorphism of the group $F_\infty\times F_\infty$ into the unitary group $U(\ell^2)$ with the strong topology is pointwise…

We construct a quasiconformal mapping of $n$-dimensional Euclidean space, $n \geq 2$, that simultaneously distorts the Hausdorff dimension of a nearly maximal collection of parallel lines by a given amount. This answers a question of…

We prove that some classes of triangle-free Artin groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin groups that are properly cubulated but cannot be…

We provide a systematic description of the automorphism groups of specially cocompact CAT(0) cube complexes. We show that these groups are topologically finitely generated, present a method to explicitly obtain generating sets, and prove a…

We prove that harmonic maps into Euclidean buildings, which are not necessarily locally finite, have singular sets of Hausdorff codimension 2, extending the locally finite regularity result of Gromov and Schoen. As an application, we prove…

For a nondegenerate additive subgroup $G$ of the $n$-dimensional vector space $F^n$ over an algebraically closed field $F$ of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type $W(G,n)$ spanned by all…

We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find new examples of such CAT(0) cube complexes…

We extend the definition of quasi-finite complexes by considering not necessarily countable complexes. We provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications.…

We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, which are special cube complexes in the sense of…

The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height…

We consider a finitely generated virtually abelian group $G$ acting properly and without inversions on a CAT(0) cube complex $X$. We prove that $G$ stabilizes a finite dimensional CAT(0) subcomplex $Y \subseteq X$ that is isometrically…

We show that an automorphism of an arbitrary CAT(0) cube complex either has a fixed point or preserves some combinatorial axis. It follows that when a group contains a distorted cyclic subgroup, it admits no proper action on a discrete…

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have…

We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and…

We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpi\'nski carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpi\'nski carpets $S_p$, $p\ge 3$ odd, we show that the…

This belongs to a series of papers motivated by Ballmann's Higher Rank Rigidity Conjecture. We prove the following. Let $X$ be a CAT(0) space with a geometric group action. Suppose that every geodesic in $X$ lies in an $n$-flat, $n\geq 2$.…

We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As…

We prove that any convex flat subset in a complete Euclidean building is contained in an apartment of the maximal system of apartments.

Ballmann's Rank Rigidity Conjecture predicts that a CAT(0) space of higher rank with a geometric group action is rigid -- isometric to a Riemannian symmetric space, a Euclidean building, or splits as a direct product. We confirm this…