arXiv research

Groups on CAT(0) cube complexes grow exponentially uniformly.

Algorithm constructs Grushko decomposition of certain groups.

We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or …

We prove that each nonpositively curved square VH-complex can be turned functorially into a locally 6-large simplicial complex of the same homotopy type. It follows that any group acting geometrically on a CAT(0) square VH-complex is systolic. In particular the product of two finitely generated free groups is systolic,…

Algorithm constructs JSJ decomposition for hyperbolic groups.

The paper presents counterexamples to LS-category conjectures and constructs maps between manifolds.

The study proves conjecture for specific Artin groups.

Our main result is that for densities $<\frac{3}{10}$ a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality, to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure o…

Locally finite complexes with polyhedral $CAT(0)$ metrics are arborescent.

Study mapping class groups on CAT(0) cube complexes.

We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex.

The study uses machine learning to predict CAT bond coupons based on climate data.

Lecture notes on CAT(0) cube complexes for beginners.

New graphs found in CAT(0) group boundaries.

We generalize the theory of gradient flows of semi-convex functions on CAT(0)-spaces, developed by Mayer and Ambrosio--Gigli--Savaré, to CAT(1)-spaces. The key tool is the so-called "commutativity" representing a Riemannian nature of the space, and all results hold true also for metric spaces satisfying the commutativi…

New group acts on complex but not in lower dimensions.

Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.

The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.

We show that the free-by-cyclic groups of the form F(2)-by-Z act properly cocompactly on CAT(0) square complexes. We also show using generalised Baumslag-Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to BS(1,j). Finally we consid…

Develops theory of relatively geometric actions on CAT(0) cube complexes.

Morse theory on complexes for CAT(0) groups.

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…

Injectivity of fundamental group map proves contractibility of intersection in CAT(0) complexes.

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) …

CAT(0) properties extended to a class of Shephard groups.

Study shows Roller compactification's median graph has limited asymptotic dimension.

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…

Asymptotically CAT(0) metrics and Z-structures for HHGs, proving Farrell-Jones Conjecture.

Study cup products on CAT(0) cube complexes, proving quasimorphisms' vanishing results.

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

Extends folding techniques to study subgroups of CAT(0) cube complexes.

Groups acting on CAT(0) cube complexes have hyperfinite boundary actions.

Groups act on cube complexes with varying dimensions.

Automorphisms of contact graphs match those of ${ m CAT(0)}$ cube complexes under weak conditions.

The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\c…

The study shows that certain complex geometries are hyperbolic and contractible but fail to be CAT(0).

Characterizes when the Roller boundary equals the Poisson boundary of CAT(0) cube complexes.

CAT(0) study of curve complements and families.

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such a…

In this article, I prove that full subcomplexes of CAT(0) simplicial 3-complexes inherit the non positive curvature condition, and describe a family of counterexamples that prove this result can not be extended to higher dimensions.

Proof shows local convexity implies global convexity in special geometric spaces.

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 applications, we provide sufficient conditions for two points in the Tits boundary t…

We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…

Homotopy equivalent boundaries of cube complexes are studied.

We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and construct such metrics on CP^2 with various orbifold structures. In particular, in relation to questions of Gromov and Davis-Moussong we construct such metrics on a compact quotient of the two-dimensional unite complex ball. In the course of th…

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 give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to pro…

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…