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3 questions
6
votes
1
answer
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Are double cosets of cyclic subgroups separable in a special linear group?
Let $A,B \in \mathrm{SL}_3(\mathbb{Z})$. Set
$$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$
Is $S$ closed in the profinite topology on
$\mathrm{SL}_3(\mathbb{...
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3
votes
1
answer
236
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Is there a bound on the rank of finite index subgroup of SL_3(Z)?
Is there an $N \in \mathbb{N}$ such that every finite index subgroup of $\mathrm{SL}_3(\mathbb{Z})$ has a generating set of size $N$?
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19
votes
2
answers
5k
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Minimal number of generators for $GL(n,\mathbb{Z})$
$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = \begin{...
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