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]. 

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. 

Is $\bar{X}^H$ contractible or at least path-connected ? 

**Background:** 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 [http://mathoverflow.net/questions/69454/finiteness-of-non-abelian-cohomology/69575#69575][1]


  [1]: http://mathoverflow.net/questions/69454/finiteness-of-non-abelian-cohomology/69575#69575