Some remarks on twin groups

TL;DR

This paper explores properties of twin groups, a class of right-angled Coxeter groups similar to braid groups, providing algorithms for equivalence and analyzing their algebraic properties such as the $R_ty$-property and co-Hopfian nature.

Contribution

It introduces an algorithm for twin group equivalence under Markov moves and establishes key algebraic properties of twin groups for the first time.

Findings

An algorithm for twin group equivalence under Markov moves.

Twin groups have the $R_ty$-property.

Twin groups are not co-Hopfian for n ≥ 3.

Abstract

The twin group $T_n$ is a right angled Coxeter group generated by $n- 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_\infty$-property and are not co-Hopfian for $n \ge 3$.