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BIRS Workshop Lecture Videos

The Congruence Subgroup Problem for Automorphism Groups Ben-Ezra, David El-Chai


In its classical setting, the Congruence Subgroup Problem (CSP) asks whether every finite index subgroup of $GL_{n}(\mathbb{Z})$ contains a principal congruence subgroup of the form \[ \ker(GL_{n}(\mathbb{Z})\to GL_{n}(\mathbb{Z}/m\mathbb{Z})) \] for some $m\in\mathbb{Z}$. It was known already in the 19th century that for $n=2$ the answer is negative, and actually $GL_{2}(\mathbb{Z})$ has many finite index subgroups which do not come from congruence considerations. On the other hand, quite surprisingly, it was proved in the sixties by Mennicke and by Bass-Lazard-Serre that for $n\geq 3$ the answer to the CSP is affirmative. This breakthrough led to a rich theory of the CSP for general arithmetic groups. $$ $$ Viewing $GL_{n}(\mathbb{Z})\cong Aut(\mathbb{Z}^{n})$ as the automorphism group of $\Gamma=\mathbb{Z}^{n}$, one can generalize the CSP to automorphism groups as follows: Let $\Gamma$ be a finitely generated group; does every finite index subgroup of $Aut(\Gamma)$ contain a principal congruence subgroup of the form \[ \ker(Aut(\Gamma)\rightarrow Aut(\Gamma/M)) \] for some finite index characteristic subgroup $M\leq\Gamma$ Considering this generalization, there are very few results when $\Gamma$ is non-abelian. For example, only in 2001 Asada proved, using concepts from Algebraic Geometry, that $Aut(F_{2})$ has an affirmative answer to the CSP, when $F_{2}$ is the free group on two generators. For $Aut(F_{n})$ when $n\geq 3$ the problem is still unsettled. $$ $$ In the talk I will give a survey of some recent results regarding the case where $\Gamma$ is non-abelian. We will see that when $\Gamma$ is a nilpotent group the CSP for $Aut(\Gamma)$ is completely determined by the CSP for arithmetic groups. We will also see that when $\Gamma$ is a finitely generated free metabelian group the picture changes and we have a dichotomy between $n=2,3$ and $n\geq 4$.

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