Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.3310

Root systems for asymmetric geometric representations of Coxeter groups

Robert G. Donnelly
View original source ↗Read source document ↗Export metadata ↓

Abstract

Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include all Kac--Moody Weyl groups. In particular, a characterization of when a non-trivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.

Explore related subjects

math.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Robert G. Donnelly. 2009-12-30. Root systems for asymmetric geometric representations of Coxeter groups. https://arxiv.org/abs/0707.3310

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Random Burnside Groups

Fix integers $m\ge2$ and $k\ge1$. For every sufficiently large fixed odd exponent $M$, choose $k$ independent elements uniformly from the sphere of radius $\ell$ in the free Burnside group $B_m(M)$. We prove that the quotient by their normal closure is infinite with probability at least $1-e^{-c\ell}$. More precisely, the quotient map has no nontrivial kernel element of length at most $κ\ell$, for constants $c,κ>0$. We also consider the analogue of the Gromov positive density model for the Burnside variety.

math.GR↗
arXiv2026

Spherical growth of reciprocal classes in the Hecke Groups

Let $r \in \mathbb{Z}$, $r>2$, and let $Γ_{2r} \cong (\mathbb{Z}/2\mathbb{Z}) \ast (\mathbb{Z}/2r\mathbb{Z})$ be the Hecke group. Let $\mathcal{N}_\ell$ denote the set of conjugacy classes of reciprocal elements with cyclically reduced word length exactly $\ell$ in the standard generating set of $Γ_{2r}$. Let $P(x)$ be the polynomial \[ P(x) = x^{r+1} - 2\sum_{i=1}^{r-1} x^{i} - 1. \] Let $ρ$ be the unique positive root of $P$ and $C$ be a constant depending on $r.$ We prove that, as $\ell \to \infty$, \[|\mathcal{N}_\ell| \sim Cρ^{\left\lfloor \tfrac{\ell+1}{2} \right\rfloor}. \] Our approach exploits the free product structure of $Γ_{2r}$, together with a combinatorial counting function and recurrence relations arising from cyclically reduced representatives. We further show that the asymptotic growth rate of primitive reciprocal conjugacy classes of word length $\ell$ coincides with that of $\mathcal{N}_\ell$. This result extends previous work for odd values of $p$ and provides a uniform asymptotic bound for all Hecke groups.

math.GR↗
arXiv2026

Weak rank rigidity for groups with a navigable path system

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective automorphisms of open convex cones, Helly groups (answering a question of Genevois), Coxeter groups, weak Garside groups (in particular Deligne's groups and fundamental groups of Salvetti complexes of oriented matroids), hierarchically hyperbolic groups, and other examples. Along the way, we show that those groups satisfy the Morse local-to-global property, providing a unified proof for the whole class. In the metric setting, the same condition of non-positive curvature allows to provide a local definition (that is, in a sense, optimal) of rank one/Morse geodesics, mirroring the one using parallel Jacobi fields from Riemannian geometry; to deduce linearity of divergence from linearity on a sequence; to obtain new cases in which Morse geodesics are strongly contracting. The main new tool introduced is the generalised contraction space, a hyperbolic space that encodes the negative curvature of a given space.

math.GR↗