arXiv research

We prove that every finite connected simplicial complex has the homology of the classifying space for some $\mathrm{CAT}(0)$ cubical duality group. More specifically, for any finite simplicial complex $X$, we construct a locally $\mathrm{CAT}(0)$ cubical complex $T_{X}$ and an acyclic map $t_{X} : T_{X} \to X$ such tha…

We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that $G$ is a group for which there is a bound on the orders of its finite subgroups. We prove that if $G$ acts properly on a finite-dimensional CAT(0) cubical complex, then either $G$ contains a free subgroup of rank 2 o…

We prove a geometric model for HHS hierarchies as CAT(0) cube complexes.

Let $Γ$ be an irreducible lattice of $\Q$-rank $\geq 2$ in a semisimple Lie group of noncompact type. We prove that any action of $Γ$ on a $\CAT(0)$ cubical complex has a global fixed point.

We prove that a hyperplane in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem is a corollary of Gromov's link condition. We also give new arguments establishing some combinato…

Survey discusses new ideas in geometric group theory and their applications.

New bicombings found for mapping class groups and Teichmüller spaces.

The contact graph of a CAT(0) cubical complex has unbounded structure and a Gaussian CLT for random walks.

New topology shows Morse boundaries are topologically invariant.

Groups act on cube complexes with varying dimensions.

Planar pure braids form a group that acts on a CAT(0) cubical complex.

We prove that an open manifold $M$ of dimension at least $5$ which admits a complete CAT(0) polyhedral metric is pseudo-collarable, its fundamental group at infinity is strongly perfectly semistable and has vanishing Chapman-Siebenmann obstruction $τ_{\infty}(M)$. Moreover, this implies that $M$ is topologically collap…

New insights into groups with uniform exponential growth.

Classifies 3-manifold groups with equivariant hierarchically hyperbolic structures.

New groups with special properties found.

We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it…

Proves hyperbolized groups are virtually compact special and linear.

We prove that a Kähler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free Abelian factor. Similarly, we prove that a closed aspherical Kähler manifold with a cu…

Extends graph factor system to quasi-median graphs.

We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of ${\rm CAT(0)}$ cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional…

Study rigidity of real moment-angle manifolds using cubical geometry.

We bound the size of $d$-dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.

We classify Dehn functions of Bestvina-Brady groups.

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups,…

In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a \emph{factor system}, and the role of the curve graph is played by the \emph{contact graph}. There are a numb…

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set, together with a pairwise-disjoint collection of subcomplexes corresponding, in the…

Proper proximality proved for various groups on non-positive curvature spaces.

We show that group actions on irreducible ${\rm CAT(0)}$ cube complexes with no free faces are uniquely determined by their $\ell^1$ length function. Actions are allowed to be non-proper and non-cocompact, as long as they are minimal and have no finite orbit in the visual boundary. This is, to our knowledge, the first …

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cu…

We study branched covering spaces in several contexts, proving that under suitable circumstances the cover satisfies the same upper curvature bounds as the base space. The first context is of a branched cover of an arbitrary metric space that satisfies Alexandrov's curvature condition CAT(k), over an arbitrary complete…

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

Toyoda proved 5-point CAT(0) spaces; this paper offers a new proof.

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group $Γ$ is said to be {\it rigid}, if $Γ$ determines the boundary up to homeomorphisms of a CAT(0) space on whi…

CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Pro…

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

Geodesics spiral around compact subsets in CAT(0) spaces.

A simpler proof shows $L^2$-metric completion is CAT(0).

We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.

In this paper, we study CAT(0) spaces with non-locally connected boundary. We give some condition of a CAT(0) space whose boundary is not locally connected.

In this paper, we study the capacity dimension of the boundary of $CAT(0)$ spaces. We first compare the two metrics on the boundary of a hyperbolic $CAT(0)$ space, i.e., the visual metric and the conical metric, and prove that they give the same capacity dimension of the boundary. Then we study the capacity dimension o…

New method connects CAT(0) spaces to hyperbolic spaces.

Study on infinite energy maps from surfaces to CAT(0) spaces.

Study of geometric actions on CAT(0) spaces and their limits.

We examine volume pinching problems of CAT(1) spaces. We characterize a class of compact geodesically complete CAT(1) spaces of small specific volume. We prove a sphere theorem for compact CAT(1) homology manifolds of small volume. We also formulate a criterion of manifold recognition for homology manifolds on volume g…

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

We study affine maps between CAT(0) spaces with geometric actions, and show that they essentially split as products of dilations and linear maps (on the Euclidean factor). This extends known results from the Riemannian case. Furthermore, we prove a splitting lemma for the Tits boundary of a CAT(0) space with geometric …

Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.