# Fixed points of the Borel-Serre compactification

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 Finiteness of non-abelian cohomology

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