Structural aspects of twin and pure twin groups

Abstract

The twin group $T_n$ is a Coxeter group generated by $n-1$ involutions and the pure twin group $P{T}_n$ is the kernel of the natural surjection of $T_n$ onto the symmetric group on $n$ letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group $T_n$ decomposes into a free product with amalgamation for $n>4$. It is shown that the pure twin group $P{T}_n$ is free for $n=3,4$, and not free for $n\ge 6$. We determine a generating set for $P{T}_n$, and give an upper bound for its rank. We also construct a natural faithful representation of $T_4$ into $\mathrm{Aut}(F_7)$. In the end, we propose virtual and welded analogues of these groups and some directions for future work.

20 pages, 7 figures, minor corrections and additions done, to appear in Geometriae Dedicata

Cited by in corpus (14)

• Topological Exchange Statistics in One Dimension
• Planar pure braids on six strands
• Structure and automorphisms of pure virtual twin groups
• Conjugacy classes and automorphisms of twin groups
• Alexander and Markov theorems for virtual doodles
• Some remarks on twin groups
• Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension
• Automorphisms of odd Coxeter groups
• A presentation for the planar pure braid group
• Congruence subgroups and crystallographic quotients of small Coxeter groups
• Virtual planar braid groups and permutations
• Topologically Nontrivial Three-Body Contact Interaction in One Dimension
• Flat braid groups, right-angled Artin groups, and commensurability
• Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups