The Finitary Andrews-Curtis Conjecture

ABSTRACT

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 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.

Famous Andrews-Curtis Conjecture from algebraic topology states that \$\Delta_k(F)\$ is connected for a free group \$F\$ of rank \$k\$. I am going to discuss the following result which can be viewed as Finitary Andrews-Curtis Conjecture:

Theorem. Let \$G\$ be a finite group and \$k \ge max{d_G(G),2}\$. Then the connected components of the AC-graph \$\Delta_k(G)\$ are precisely the preimages of the connected components of the AC-graph of the abelianization \$\Delta_k(G/[G,G])\$.

This is joint work with Alexandre Borovik and Alex Lubotzky.

Speaker's Contact Info: alexei(at-sign)rio.sci.ccny.cuny.edu