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 The Congruence Subgroup Problem for Automorphism Groups
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The Congruence Subgroup Problem for Automorphism Groups BenEzra, David ElChai
Description
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 BassLazardSerre 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 nonabelian. 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 nonabelian. 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 $n\geq 4$.
Item Metadata
Title 
The Congruence Subgroup Problem for Automorphism Groups

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20200909T13:04

Description 
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 BassLazardSerre 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
nonabelian. 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 nonabelian. 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 $n\geq 4$.

Extent 
29.0 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: The Hebrew University

Series  
Date Available 
20210309

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0396073

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Postdoctoral

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International