arXiv research

Study on connectivity of Morse boundaries of Coxeter groups.

We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of dia…

Let $Γ$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $Γ$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_Γ$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometr…

Proves involutions on Right-angled Coxeter groups without fixed points.

The paper proves conditions for the isomorphism between standard and dual Artin groups.

We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle…

New Coxeter groups have unique boundary structures.

The paper shows symmetries of a geometric space for Coxeter groups.

Using the Guirardel-Levitt outer space of a free product, we prove that the outer automorphism group of the outer automorphism group of the universal Coxeter group of rank $n \geq 5$ is trivial, and that it is a cyclic group of order 2 if $n=4$. In addition we prove that the outer automorphism group of the automorphism…

In the present paper, we prove that the group $G_{n}^{2}$ of free $n$-strand braids is isomorphic to a subgroup of a semidirect product of some Coxeter group that we denote by $C(n,2)$ and the symmetric group $S_{n}$.

The paper shows that for Coxeter groups, the commensurator of outer automorphisms is rigid.

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abel…

Proves almost profinite rigidity for certain free-by-cyclic groups.

In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f…

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

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-end…

In this paper we show how to obtain representations of Coxeter groups acting on H^n to certain classical groups. We determine when the kernel of such a representation is torsion-free and thus the quotient a hyperbolic n-manifold.

This article announces joint work with Frank Connolly and Jim Davis. We generalize our classification of pseudo-free involutions on the n-torus, by studying the action of the associated infinite group with torsion in the universal cover. Included is a non-Riemannian example obtained from the restriction of the action o…

We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer …

Study on stable commutator length in RAAGs and Coxeter groups, proving spectral gaps and hardness results.

Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös-Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about t…

Survey on Coxeter groups for Lie group examples.

We extend a systolic inequality of Guth for Riemannian manifolds of maximal $\mathbb{Z}_2$ cup-length to piecewise Riemannian complexes of dimension 2. As a consequence we improve the previous best universal lower bound for the systolic area of groups for a large class of groups, including free abelian and surface grou…

New proof shows not all Salem numbers are growth rates of Coxeter groups.

Characterizes Coxeter groups with convex cocompact representations in projective space.

By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction of hyperbolic manifolds of very small volume in up to 8 dimensions.

We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.

New proofs for growth series of Coxeter groups using complex structures.

New groups found in hyperbolic 4D and 5D space have minimal growth rate.

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 consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for c…

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-…

Flat coordinates found for algebraic Frobenius manifolds in low dimensions.

Surprising circles found in Coxeter group boundaries.

We prove that two angle-compatible Coxeter generating sets of a given finitely generated Coxeter group are conjugate provided one of them does not admit any elementary twist. This confirms a basic case of a general conjecture which describes a potential solution to the isomorphism problem for Coxeter groups.

New method constructs non-quasiconvex subgroups in hyperbolic groups.

Study Coxeter groups over fusion rings and their geometric realisations.

Two quandles from Coxeter groups studied, showing similarities in automorphism groups.

We study homomorphisms from Kähler groups to Coxeter groups. As an application, we prove that a cocompact complex hyperbolic lattice (in complex dimension at least 2) does not embedd into a Coxeter group or a right-angled Artin group. This is in contrast with the case of real hyperbolic lattices.

Locally connected boundaries of Coxeter groups are studied.

Characterizes Coxeter groups with specific boundary shapes.

We study relatively hyperbolic Coxeter groups of type $HM$ with maximal Euclidean Coxeter subgroups of codimension 1. Our main result in this paper is that the dimension of these groups is bounded above.

For right-angled Coxeter groups $W_Γ$, we obtain a condition on $Γ$ that is necessary and sufficient to ensure that $W_Γ$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, …

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 f…

This paper introduces cluster exchange groupoids for Coxeter-Dynkin diagrams and finds their fundamental groups are braid groups.

In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter g…

Study on divergence and thickness for Coxeter groups, generalizing previous work.

The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…