Maxap groups with zero-dimensional group compactificationsResidually finiteness for a factor groupA free group is residually nilpotentResidually Finite Braid GroupExtensions of group homomorphisms for special groupsTopological/geometric interpretation of conjugacy separabilityhomomorphisms between free abelian groups that are not finitely generatedA consequence of $beta Xsetminus X$ not being zero dimensional.Free groups are residually of rank 2If the inner automorphism group of a group $G$ is residually finite, then $G$ is residually finite group.Is the free product of residually finite groups always residually finite?
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Maxap groups with zero-dimensional group compactifications
Residually finiteness for a factor groupA free group is residually nilpotentResidually Finite Braid GroupExtensions of group homomorphisms for special groupsTopological/geometric interpretation of conjugacy separabilityhomomorphisms between free abelian groups that are not finitely generatedA consequence of $beta Xsetminus X$ not being zero dimensional.Free groups are residually of rank 2If the inner automorphism group of a group $G$ is residually finite, then $G$ is residually finite group.Is the free product of residually finite groups always residually finite?
$begingroup$
Suppose $G$ is a countable group. We say that $G$ is maxap if there is an injective homomorphism $phi: Gto K$ with $K$ a compact group. My question is what we can demand of $K$. Given $G$ a maxap group, can we demand that $K$ be zero-dimensional? Certainly this is true when $G$ is residually finite but in general it seems tricky.
group-theory compactification
asked Mar 22 at 15:16
AndyAndy
1365
$endgroup$
add a comment |
$begingroup$
Suppose $G$ is a countable group. We say that $G$ is maxap if there is an injective homomorphism $phi: Gto K$ with $K$ a compact group. My question is what we can demand of $K$. Given $G$ a maxap group, can we demand that $K$ be zero-dimensional? Certainly this is true when $G$ is residually finite but in general it seems tricky.
group-theory compactification
asked Mar 22 at 15:16
AndyAndy
1365
$endgroup$
add a comment |
$begingroup$
Suppose $G$ is a countable group. We say that $G$ is maxap if there is an injective homomorphism $phi: Gto K$ with $K$ a compact group. My question is what we can demand of $K$. Given $G$ a maxap group, can we demand that $K$ be zero-dimensional? Certainly this is true when $G$ is residually finite but in general it seems tricky.
group-theory compactification
asked Mar 22 at 15:16
AndyAndy
1365
$endgroup$
Suppose $G$ is a countable group. We say that $G$ is maxap if there is an injective homomorphism $phi: Gto K$ with $K$ a compact group. My question is what we can demand of $K$. Given $G$ a maxap group, can we demand that $K$ be zero-dimensional? Certainly this is true when $G$ is residually finite but in general it seems tricky.
group-theory compactification
group-theory compactification
asked Mar 22 at 15:16
AndyAndy
1365
asked Mar 22 at 15:16
AndyAndy
1365
asked Mar 22 at 15:16
AndyAndy
1365
asked Mar 22 at 15:16
AndyAndy
1365
asked Mar 22 at 15:16
AndyAndy
1365
1365
add a comment |
add a comment |
1 Answer
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$begingroup$
Precisely, this is possible iff $G$ is residually finite (because 0-dimensional compact groups are profinite, hence residually finite).
So $mathbfQ$ is an example of maximally almost periodic with no injective homomorphism into such a $K$.
(For finitely generated groups however, maxap $Leftrightarrow$ residually finite.)
answered Mar 22 at 15:24
YCorYCor
8,5171129
$endgroup$
add a comment |
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$begingroup$
Precisely, this is possible iff $G$ is residually finite (because 0-dimensional compact groups are profinite, hence residually finite).
So $mathbfQ$ is an example of maximally almost periodic with no injective homomorphism into such a $K$.
(For finitely generated groups however, maxap $Leftrightarrow$ residually finite.)
answered Mar 22 at 15:24
YCorYCor
8,5171129
$endgroup$
add a comment |
$begingroup$
Precisely, this is possible iff $G$ is residually finite (because 0-dimensional compact groups are profinite, hence residually finite).
So $mathbfQ$ is an example of maximally almost periodic with no injective homomorphism into such a $K$.
(For finitely generated groups however, maxap $Leftrightarrow$ residually finite.)
answered Mar 22 at 15:24
YCorYCor
8,5171129
$endgroup$
add a comment |
$begingroup$
Precisely, this is possible iff $G$ is residually finite (because 0-dimensional compact groups are profinite, hence residually finite).
So $mathbfQ$ is an example of maximally almost periodic with no injective homomorphism into such a $K$.
(For finitely generated groups however, maxap $Leftrightarrow$ residually finite.)
answered Mar 22 at 15:24
YCorYCor
8,5171129
$endgroup$
Precisely, this is possible iff $G$ is residually finite (because 0-dimensional compact groups are profinite, hence residually finite).
So $mathbfQ$ is an example of maximally almost periodic with no injective homomorphism into such a $K$.
(For finitely generated groups however, maxap $Leftrightarrow$ residually finite.)
answered Mar 22 at 15:24
YCorYCor
8,5171129
answered Mar 22 at 15:24
YCorYCor
8,5171129
answered Mar 22 at 15:24
YCorYCor
8,5171129
answered Mar 22 at 15:24
YCorYCor
8,5171129
8,5171129
add a comment |
add a comment |
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