Curve complexes and Garside groups

TL;DR

This paper introduces a new metric space called the additional length complex for Garside groups, sharing properties with curve complexes, and proves key conjectures for braid groups, linking algebraic and geometric group theory.

Contribution

It constructs the additional length complex for Garside groups, proves its infinite diameter for braid groups, and explores the actions of different braid types within this space.

Findings

The additional length complex is $ ext{delta}$-hyperbolic and typically infinite.

Reducible and periodic braids act elliptically, pseudo-Anosov braids act loxodromically.

The complex is conjectured to be quasi-isometric to the curve complex of the punctured disk.

Abstract

We present a simple construction which associates to every Garside group a metric space, called the additional length complex, on which the group acts. These spaces share important features with curve complexes: they are $\delta$-hyperbolic, infinite, and typically locally infinite graphs. We conjecture that, apart from obvious counterexamples, additional length complexes are always of infinite diameter. We prove this conjecture for the classical example of braid groups $(B_n,\Delta)$; moreover, in this framework, reducible and periodic braids act elliptically, and at least some pseudo-Anosov braids act loxodromically. We conjecture that for $B_n$, the additional length complex is actually quasi-isometric to the curve complex of the $n$ times punctured disk.