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







share|cite|improve this question











$endgroup$








  • 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
















0












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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














0












0








0





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 21 at 1:35







suchan

















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














  • 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










1 Answer
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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 $J$.






share|cite|improve this answer









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    0












    $begingroup$

    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 $J$.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      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 $J$.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        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 $J$.






        share|cite|improve this answer









        $endgroup$



        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 $J$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 22 at 2:04









        Moishe CohenMoishe Cohen

        47.4k343108




        47.4k343108






























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