Mixing
Characterization or examples of metric spaces with this property
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Let $X$ be a metric space and let $J$ be a subgroup of $text{Isom}(X)$. For any $x in X$ and compact subset $K subset X$, consider the set
$$A = left{ g in J : g(x) in K right}.$$
What conditions on $X$ and/or J are necessary/sufficient in order to conclude that $A subset text{Isom}(X)$ is compact? (Here $text{Isom}(X)$ carries the compact-open topology).
For example, I know $A$ is compact when $X = mathbb{H}^2$ and $J = text{SL}(2,mathbb{R})$ or $J = text{PSL}(2,mathbb{R})$.
general-topology geometric-group-theory
asked Jan 17 at 21:18
suchansuchan
217110
$endgroup$
add a comment |
$begingroup$
Let $X$ be a metric space and let $J$ be a subgroup of $text{Isom}(X)$. For any $x in X$ and compact subset $K subset X$, consider the set
$$A = left{ g in J : g(x) in K right}.$$
What conditions on $X$ and/or J are necessary/sufficient in order to conclude that $A subset text{Isom}(X)$ is compact? (Here $text{Isom}(X)$ carries the compact-open topology).
For example, I know $A$ is compact when $X = mathbb{H}^2$ and $J = text{SL}(2,mathbb{R})$ or $J = text{PSL}(2,mathbb{R})$.
general-topology geometric-group-theory
asked Jan 17 at 21:18
suchansuchan
217110
$endgroup$
1
$begingroup$
Do you know the statement of Arzela-Ascoli theorem?
$endgroup$
– Moishe Cohen
Jan 17 at 21:20
1
$begingroup$
You need to first endow $mathrm{Isom}(X)$ with a topology for the question to make sense.
$endgroup$
– YCor
Jan 18 at 13:01
add a comment |
$begingroup$
Let $X$ be a metric space and let $J$ be a subgroup of $text{Isom}(X)$. For any $x in X$ and compact subset $K subset X$, consider the set
$$A = left{ g in J : g(x) in K right}.$$
What conditions on $X$ and/or J are necessary/sufficient in order to conclude that $A subset text{Isom}(X)$ is compact? (Here $text{Isom}(X)$ carries the compact-open topology).
For example, I know $A$ is compact when $X = mathbb{H}^2$ and $J = text{SL}(2,mathbb{R})$ or $J = text{PSL}(2,mathbb{R})$.
general-topology geometric-group-theory
asked Jan 17 at 21:18
suchansuchan
217110
$endgroup$
Let $X$ be a metric space and let $J$ be a subgroup of $text{Isom}(X)$. For any $x in X$ and compact subset $K subset X$, consider the set
$$A = left{ g in J : g(x) in K right}.$$
What conditions on $X$ and/or J are necessary/sufficient in order to conclude that $A subset text{Isom}(X)$ is compact? (Here $text{Isom}(X)$ carries the compact-open topology).
For example, I know $A$ is compact when $X = mathbb{H}^2$ and $J = text{SL}(2,mathbb{R})$ or $J = text{PSL}(2,mathbb{R})$.
general-topology geometric-group-theory
general-topology geometric-group-theory
asked Jan 17 at 21:18
suchansuchan
217110
asked Jan 17 at 21:18
suchansuchan
217110
edited Jan 21 at 1:35
suchan
asked Jan 17 at 21:18
suchansuchan
217110
asked Jan 17 at 21:18
suchansuchan
217110
asked Jan 17 at 21:18
suchansuchan
217110
217110
1
$begingroup$
Do you know the statement of Arzela-Ascoli theorem?
$endgroup$
– Moishe Cohen
Jan 17 at 21:20
1
$begingroup$
You need to first endow $mathrm{Isom}(X)$ with a topology for the question to make sense.
$endgroup$
– YCor
Jan 18 at 13:01
add a comment |
1
$begingroup$
Do you know the statement of Arzela-Ascoli theorem?
$endgroup$
– Moishe Cohen
Jan 17 at 21:20
1
$begingroup$
You need to first endow $mathrm{Isom}(X)$ with a topology for the question to make sense.
$endgroup$
– YCor
Jan 18 at 13:01
1
1
$begingroup$
Do you know the statement of Arzela-Ascoli theorem?
$endgroup$
– Moishe Cohen
Jan 17 at 21:20
$begingroup$
Do you know the statement of Arzela-Ascoli theorem?
$endgroup$
– Moishe Cohen
Jan 17 at 21:20
1
1
$begingroup$
You need to first endow $mathrm{Isom}(X)$ with a topology for the question to make sense.
$endgroup$
– YCor
Jan 18 at 13:01
$begingroup$
You need to first endow $mathrm{Isom}(X)$ with a topology for the question to make sense.
$endgroup$
– YCor
Jan 18 at 13:01
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Given your examples, I assume that you are mostly interested in proper metric spaces, i.e. metric spaces where a subset if compact if and only if it is closed and bounded. The most natural topology on the group of isometries $Isom(X)$ in this case is the topology of uniform convergence on compact subsets. (Equivalently, you can equip $Isom(X)$ with topology of pointwise convergence.)
You probably also want to consider closed subgroups $J< Isom(X)$.
It is an application of Arzela-Ascoli theorem (combined with the diagonal argument) that in this setting the subset
$$
A_{J,K,x} = left{ g in J : g(x) in K right}$$
is compact in $Isom(X)$ and, hence (since $J$ is closed), in $J$.
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Given your examples, I assume that you are mostly interested in proper metric spaces, i.e. metric spaces where a subset if compact if and only if it is closed and bounded. The most natural topology on the group of isometries $Isom(X)$ in this case is the topology of uniform convergence on compact subsets. (Equivalently, you can equip $Isom(X)$ with topology of pointwise convergence.)
You probably also want to consider closed subgroups $J< Isom(X)$.
It is an application of Arzela-Ascoli theorem (combined with the diagonal argument) that in this setting the subset
$$
A_{J,K,x} = left{ g in J : g(x) in K right}$$
is compact in $Isom(X)$ and, hence (since $J$ is closed), in $J$.
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
$endgroup$
add a comment |
$begingroup$
Given your examples, I assume that you are mostly interested in proper metric spaces, i.e. metric spaces where a subset if compact if and only if it is closed and bounded. The most natural topology on the group of isometries $Isom(X)$ in this case is the topology of uniform convergence on compact subsets. (Equivalently, you can equip $Isom(X)$ with topology of pointwise convergence.)
You probably also want to consider closed subgroups $J< Isom(X)$.
It is an application of Arzela-Ascoli theorem (combined with the diagonal argument) that in this setting the subset
$$
A_{J,K,x} = left{ g in J : g(x) in K right}$$
is compact in $Isom(X)$ and, hence (since $J$ is closed), in $J$.
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
$endgroup$
add a comment |
$begingroup$
Given your examples, I assume that you are mostly interested in proper metric spaces, i.e. metric spaces where a subset if compact if and only if it is closed and bounded. The most natural topology on the group of isometries $Isom(X)$ in this case is the topology of uniform convergence on compact subsets. (Equivalently, you can equip $Isom(X)$ with topology of pointwise convergence.)
You probably also want to consider closed subgroups $J< Isom(X)$.
It is an application of Arzela-Ascoli theorem (combined with the diagonal argument) that in this setting the subset
$$
A_{J,K,x} = left{ g in J : g(x) in K right}$$
is compact in $Isom(X)$ and, hence (since $J$ is closed), in $J$.
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
$endgroup$
Given your examples, I assume that you are mostly interested in proper metric spaces, i.e. metric spaces where a subset if compact if and only if it is closed and bounded. The most natural topology on the group of isometries $Isom(X)$ in this case is the topology of uniform convergence on compact subsets. (Equivalently, you can equip $Isom(X)$ with topology of pointwise convergence.)
You probably also want to consider closed subgroups $J< Isom(X)$.
It is an application of Arzela-Ascoli theorem (combined with the diagonal argument) that in this setting the subset
$$
A_{J,K,x} = left{ g in J : g(x) in K right}$$
is compact in $Isom(X)$ and, hence (since $J$ is closed), in $J$.
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
answered Jan 22 at 2:04
Moishe CohenMoishe Cohen
47.4k343108
47.4k343108
add a comment |
add a comment |
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$begingroup$
Do you know the statement of Arzela-Ascoli theorem?
$endgroup$
– Moishe Cohen
Jan 17 at 21:20
1
$begingroup$
You need to first endow $mathrm{Isom}(X)$ with a topology for the question to make sense.
$endgroup$
– YCor
Jan 18 at 13:01