Discrete subgroups of isometry group of proper metric space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:28:57Z http://mathoverflow.net/feeds/question/90439 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90439/discrete-subgroups-of-isometry-group-of-proper-metric-space Discrete subgroups of isometry group of proper metric space unknown 2012-03-07T09:58:27Z 2012-03-07T12:51:42Z <p>Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$.</p> <p>Consider the following conditions on $G$:</p> <p>(1) $G$ acts properly on $X$, i.e. any two points $x$ and $y$ in $X$ have neighborhoods $U_x$ and $U_y$ such that there are only a finite number of group elements $g \in G$ with $g(U_x)$ meeting $U_y$.</p> <p>(2) $G$ is a discrete subgroup of $\mathrm{ISO}(X)$.</p> <p>(3) The orbit $Gx$ is a discrete subset of $X$ for all $x \in X$.</p> <p>My question: Is (1) equivalent with (2) or is (2) equivalent with (3), or neither? Does anything change if one assumes also that $X$ is CAT(0) and/or $G$ acts cocompactly. </p> http://mathoverflow.net/questions/90439/discrete-subgroups-of-isometry-group-of-proper-metric-space/90449#90449 Answer by Misha for Discrete subgroups of isometry group of proper metric space Misha 2012-03-07T12:51:42Z 2012-03-07T12:51:42Z <p>This is a nice homework problem for a graduate class in metric geometry or geometric group theory right after the students learned the definition of uniform convergence on compacts and the Arzela-Ascoli theorem. (No, the students do not need to know what CAT(0) spaces are.) </p>