Montreal Geometric & Combinatorial Group Theory Seminar

Speaker:   Alexei Miasnikov (McGill)
Title:         “The Finitary Andrews-Curtis Conjecture”
Date:        3:30PM, Wednesday, November 5, 2003
Place:       Room 920, Burnside Hall, McGill University

Abstract:

(joint with Alexandre Borovik and Alex Lubotzky)

Let $G$ be a group, $d_G(G)$ the minimal number of generators of

$G$ as a normal subgroup, $k \geq d_G(G)$, and $N_k(G)$ the set of

all $k$-tuples of elements in $G$ which generate $G$ as a normal

subgroup. Then the {\em Andrews--Curtis graph} $\Delta_k(G)$ of

the group $G$ is the graph whose vertices are tuples from

$N_k(G)$ and such that two tuples are connected by an edge if one

of them is obtained from another by an elementary Nielsen

transformation or by a conjugation of one of the components of the

tuple. Two tuples $U, V \in N_k(G)$ are AC-equivalent if they

belong to the same connected component of $\Delta_k(G)$.

Famous Andrews-Curtis Conjecture from algebraic topology asks

whether the graph $\Delta_k(F)$ is connected for a free group

$F$ of rank $k$.

It is known that the Andrews-Curtis graph $\Delta_{k}(G)$ is not

connected in general (there are counter examples in abelian groups

$G$ for $k = d_G(G)$).

\noindent {\bf Theorem} {\bf ({\em Finitary Andrews-Curtis

Conjecture})} {\em Let $G$ be a finite group and $k \geq

\max\{d_G(G),2\}$. Then two tuples $U, V$ from $N_k(G)$ are

AC-equivalent if and only if they are AC-equivalent in the

abelianization ${\rm Ab}(G) = G/[G,G]$, i.e., the connected

components of the AC-graph $\Delta_k(G)$ are precisely the

preimages of the connected components of the AC-graph

$\Delta_k({\rm Ab}(G))$.}