Fixed points of the Borel-Serre compactification - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T12:03:24Z http://mathoverflow.net/feeds/question/70659 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70659/fixed-points-of-the-borel-serre-compactification Fixed points of the Borel-Serre compactification Ralph 2011-07-18T19:03:25Z 2011-07-18T19:03:25Z <p>Let $\Gamma$ be an arithmetic group and $X$ its symmetric space. Borel-Serre constructed a space $\bar{X} \supset X$ such that $\bar{X}/\Gamma$ is a compactification of $X/\Gamma$ [Corners and Arithmetic Groups, Comm. Math. Helv. 48(1973), 436-491, §7]. </p> <p>Moreover $\bar{X}$ is a contractible, finite-dimensional CW-complex and $\Gamma$ operates properly and cellularly on $\bar{X}$. In particular, if $H \le \Gamma$ is a finite subgroup, then the fixed point space $\bar{X}^H$ is non-empty. </p> <p>Is $\bar{X}^H$ contractible or at least path-connected ? </p> <p><strong>Background:</strong> If so, it would follow that the non-abelian cohomology $H^1(G;\Gamma)$ is finite for $\Gamma$ arithmetic and $G \subseteq \operatorname{Aut}(\Gamma)$ finite. See also <a href="http://mathoverflow.net/questions/69454/finiteness-of-non-abelian-cohomology/69575#69575" rel="nofollow">http://mathoverflow.net/questions/69454/finiteness-of-non-abelian-cohomology/69575#69575</a></p>