Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2010.13145

Euclidean Artin-Tits groups are acylindrically hyperbolic

Matthieu Calvez
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we show the statement in the title. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the \emph{additional length graph} and they used it to show that central quotients of Artin-Tits groups of spherical type are acylindrically hyperbolic. In general, a euclidean Artin-Tits group is not \emph{a priori} a Garside group but McCammond and Sulway have shown that it embeds into an \emph{infinite-type} Garside group which they call a \emph{crystallographic Garside group}. We associate a \emph{hyperbolic} additional length graph to this crystallographic Garside group and we exhibit elements of the euclidean Artin-Tits group which act loxodromically and WPD on this hyperbolic graph.

Explore related subjects

math.GR

Keep this discovery

BibTeXRIS

Matthieu Calvez. 2020-10-25. Euclidean Artin-Tits groups are acylindrically hyperbolic. https://arxiv.org/abs/2010.13145

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

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗