2-dimensional Shephard groups

TL;DR

This paper investigates 2-dimensional Shephard groups, showing their geometric properties, hyperbolicity, and residual finiteness, by constructing specific complexes and analyzing curvature conditions.

Contribution

It introduces a CAT(0) complex for 2D Shephard groups and establishes their acylindrical and relative hyperbolicity, extending known properties from Artin groups.

Findings

Large powers in quotients prevent CAT(0) structure.

Constructed a CAT(0) complex for these groups.

Proved acylindrical and relative hyperbolicity.

Abstract

The 2-dimensional Shephard groups are quotients of 2-dimensional Artin groups by powers of standard generators. We show that such a quotient is not $\mathrm{CAT}(0)$ if the powers taken are sufficiently large. However, for a given 2-dimensional Shephard group, we construct a $\mathrm{CAT}(0)$ piecewise Euclidean cell complex with a cocompact action (analogous to the Deligne complex for an Artin group) that allows us to determine other non-positive curvature properties. Namely, we show the 2-dimensional Shephard groups are acylindrically hyperbolic (which was known for 2-dimensional Artin groups), and relatively hyperbolic (which most Artin groups are known not to be). As an application, we show that a broad class of 2-dimensional Artin groups are residually finite.