Montreal Geometric & Combinatorial Group Theory Seminar

Speaker:   Denis Serbin (McGill)
Title:         “Infinite words and length functions
Date:        3:30PM, Wednesday, October 29, 2003
Place:       Room 920, Burnside Hall, McGill University

Let $F=F(X)$ be a free group with basis $X$ and $\mathbb{Z}[t]$ be a ring of

polynomials in variable $t$ with integer coefficients. At first we show how to

represent elements of Lyndon's free $\mathbb{Z}[t]$-group $F^{\mathbb{Z}[t]}$

by infinite words defined as sequences  $w: [1,f_w] \rightarrow X^{\pm 1}$  over closed

intervals $[1,f_w], f_w \geq 0,$ in the additive group $\mathbb{Z}[t]^+$, viewed

as an ordered abelian group. This representation provides a natural regular free

Lyndon length function $w \rightarrow f_w$ on $F^{\mathbb{Z}[t]}$ with values

in $\mathbb{Z}[t]^+$. The second part of the talk is concerned with applications of the

construction above to finitely generated subgroups of $F^{\mathbb{Z}[t]}$. Finitely

generated subgroups of $F^{\mathbb{Z}[t]}$ are associated with combinatorial objects

called $(\mathbb{Z}[t],X)$-graphs study of which solves some algorithmic

problems for these subgroups such as the membership problem, the conjugacy problem etc. 

This is joint work with Alexei Miasnikov and Vladimir Remeslennikov.