On Shephard Groups with Large Triangles

TL;DR

This paper demonstrates that an infinite family of Shephard groups, which are extensions of Artin and Coxeter groups and not part of either class, are bi-automatic using small cancellation theory.

Contribution

It introduces a new class of Shephard groups and proves their bi-automaticity, expanding understanding of their algebraic properties.

Findings

The identified Shephard groups are bi-automatic.

The groups are not Artin or Coxeter groups.

Small cancellation techniques are effective for these groups.

Abstract

Shephard groups are common extensions of Artin and Coxeter groups. They appear, for example, in algebraic study of manifolds. An infinite family of Shephard groups which are not Artin or Coxeter groups is considered. Using techniques form small cancellation theory we show that the groups in this family are bi-automatic.