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

Abstract:

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.