# Lefschetz fixed point formula for arithmetic quotients in presence of torsion

Hey everyone,

Is there any Lefschetz fixed point formula for arithmetic quotients $X / \Gamma$ where $\Gamma$ is not necessarily torsion free? More precisely, let $G$ be a semisimple Lie group with associated symmetric space $X$. Let $\Gamma$ be an arithmetic subgroup of $G$, which is not necessarily torsion free. Let $V$ be a locally constant sheaf (induced from finite dimensional complex representation of G) on $X/\Gamma$ and $\tau$ be an involution acting on $X$, $\Gamma$ and $V$ in a compatible way. Then, $\tau$ induces an involution on the cohomology spaces $H^i(X/\Gamma,V)$ and the cohomological dimension of $X/\Gamma$ is finite. So, we can define the Lefschetz number $$L:= L(\tau,X/\Gamma,V) =\sum_{i=1}^n (-1)^i\ {\rm Tr}(\tau \mid H^i(X/\Gamma,V)).$$ Is there any formula giving $L$ in terms of the Lefschetz number of the set of fixed points of $\tau$ on $X/\Gamma$?

There are such formulas when $\Gamma$ is torsion free. I am especially interested in the case that $\Gamma$ has torsion.

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I think this is in the thesis of Juergen Blume, see books.google.com/books/about/… unfortunately, it is in German and was never published in english. If I was to suggest a strategy for a proof, I would say choose a normal subgroup $\Sigma$ of finite index which is torsion-free and decompose all terms in the ensuing spectral sequence with respect to the action of the finite group $\Gamma/\Sigma$. –  doug Sep 15 '11 at 20:02
Many thanks! I will try to get the thesis. -Seyfi Turkelli –  Turkelli Sep 16 '11 at 0:38