Some remarks on twin groups

Tushar Kanta Naik, Neha Nanda, Mahender Singh

Published 2019-12-03Version 1

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. More precisely, we show that twin groups $T_n$ have $R_{\infty }$-property and are not co-Hopfian for $n \geq 3$. We also give an algorithm for two twins to be equivalent under individual Markov moves.