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The Congruence Subgroup Problem for Automorphism Groups

Banff International Research Station for Mathematical Innovation and Discovery

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 $n\geq 4$.

Mathematics; Group theory and generalizations; Algebraic geometry; Algebraic structures

video/mp4

Author affiliation: The Hebrew University

2021-03-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77484

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/