The Finitary Andrews-Curtis Conjecture

Alexei Myasnikov

City College, CUNY

May 14,
refreshments at 3:45pm


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)

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