arXiv subjects

Ivan Levcovitz

Publications and source records attributed to Ivan Levcovitz.

12 recordsLinked to original sources

arXiv2025

Connectivity of Coxeter group Morse boundaries

We study the connectivity of Morse boundaries of Coxeter groups. We define two conditions on the defining graph of a Coxeter group: wide-avoidant and wide-spherical-avoidant. We show that wide-spherical-avoidant, one-ended, affine-free Coxeter groups have connected and locally connected Morse boundaries. On the other hand, one-ended Coxeter groups that are not wide-avoidant and not wide have disconnected Morse boundary. For the right-angled case, we get a full characterization: a one-ended right-angled Coxeter group has connected, non-empty Morse boundary if and only if it is wide-avoidant. Along the way we characterize Morse geodesic rays in affine-free Coxeter groups as those that spend uniformly bounded time in cosets of wide special subgroups.

math.GR↗

arXiv2022

Coarse cubical rigidity

We show that for many right-angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: they induce the same coarse median structure on the group. These are the first examples of non-hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank $\geq 3$, we show that all automorphism preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo-Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation.

math.GR↗

arXiv2022

Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques

We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are as short as possible (length two words), finitely generated free subgroups, non-finitely presentable subgroups, and subgroups of fundamental groups of square complexes of nonpositive sectional curvature.

math.GR↗

arXiv2022

Characterizing divergence and thickness in right-angled Coxeter groups

We completely classify the possible divergence functions for right-angled Coxeter groups (RACGs). In particular, we show that the divergence of any such group is either polynomial, exponential or infinite. We prove that a RACG is strongly thick of order k if and only if its divergence function is a polynomial of degree k+1. Moreover, we show that the exact divergence function of a RACG can easily be computed from its defining graph by an invariant we call the hypergraph index.

math.GT↗

arXiv2022

Counting lattices in products of trees

A BMW group of degree $(m,n)$ is a group that acts simply transitively on vertices of the product of two regular trees of degrees $m$ and $n$. We show that the number of commensurability classes of BMW groups of degree $(m,n)$ is bounded between $(mn)^{αmn}$ and $(mn)^{βmn}$ for some $0<α<β$. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree $(m,n)$ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.

math.GR↗

arXiv2021

Subgroups of right-angled Coxeter groups via Stallings-like techniques

We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-ended Coxeter subgroups of a 2-dimensional RACG. We provide an algorithm that determines whether a given one-ended, 2-dimensional RACG is isomorphic to some finite-index subgroup of another given RACG. In addition, we answer several algorithmic questions regarding quasiconvex subgroups. Finally, we give a new proof of Haglund's result that quasiconvex subgroups of RACGs are separable.

math.GT↗

arXiv2021

Comparing the Roller and B(X) boundaries of CAT(0) cube complexes

The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X is locally finite, essential, irreducible, non-Euclidean and admits a cocompact action by a group G, Nevo-Sageev show that a subset, B(X), of the Roller boundary is the realization of the Poisson boundary and that the action of G on B(X) is minimal and strongly proximal. Additionally, these authors show B(X) satisfies many other desirable dynamical and topological properties. In this article we give several equivalent characterizations for when B(X) is equal to the entire Roller boundary. As an application we show, under mild hypotheses, that if X is also 2-dimensional then X is G-equivariantly quasi-isometric to a CAT(0) cube complex X' whose Roller boundary is equal to B(X'). Additionally, we use our characterization to show that the usual CAT(0) cube complex for which an infinite right-angled Coxeter/Artin group acts on geometrically has Roller boundary equal to B(X), as long as the corresponding group does not decompose as a direct product.

math.GT↗

arXiv2020

Planar lattice subsets with minimal vertex boundary

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We characterize uniquely minimal sets of X: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of X: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph G of minimal sets whose vertices are congruence classes of minimal sets of X and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.

math.CO↗

arXiv2020

Right-angled Artin subgroups of right-angled Coxeter and Artin groups

We determine when certain natural classes of subgroups of right-angled Coxeter groups (RACGs) and right-angled Artin groups (RAAGs) are themselves RAAGs. We characterize finite-index visual RAAG subgroups of 2-dimensional RACGs. As an application, we show that any 2-dimensional, one-ended RACG with planar defining graph is quasi-isometric to a RAAG if and only if it is commensurable to a RAAG. Additionally, we give new examples of RACGs with non-planar defining graphs which are commensurable to RAAGs. Finally, we give a new proof of a result of Dyer: every subgroup generated by conjugates of RAAG generators is itself a RAAG.

math.GR↗

arXiv2019

Thick groups have trivial Floyd boundary

We prove that thick groups (and more generally thick graphs) have trivial Floyd boundary. This shows a wide class of finitely generated groups that are non-relatively hyperbolic have trivial Floyd boundary. In addition to giving new examples, our result provides a common proof and framework for many of the known results in the literature.

math.GT↗

arXiv2018

A quasi-isometry invariant and thickness bounds for right-angled Coxeter groups

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group's defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group's order of thickness, order of algebraic thickness and divergence function. Finally, given an integer n>1, we give examples of right-angled Coxeter groups which are thick of order n, yet are algebraically thick of order strictly larger than n, answering a question of Behrstock-Drutu-Mosher.

math.GT↗

arXiv2018

Divergence of CAT(0) Cube Complexes and Coxeter Groups

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 cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. As another application, we provide an inductive graph theoretic criteria on a right-angled Coxeter group's defining graph which allows us to recognize arbitrary integer degree polynomial divergence for many infinite classes of right-angled Coxeter groups. We also provide similar divergence results for some classes of Coxeter groups which are not right-angled.

math.GT↗