most recent 30 from http://mathoverflow.net 2013-06-18T22:41:09Z http://mathoverflow.net/feeds/question/75518 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75518/lefschetz-fixed-point-formula-for-arithmetic-quotients-in-presence-of-torsion Turkelli 2011-09-15T14:02:42Z 2011-09-15T14:02:42Z

<p>Hey everyone,</p> <p>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 $X/\Gamma$? </p> <p>There are such formulas when $\Gamma$ is torsion free. I am especially interested in the case that $\Gamma$ has torsion. </p>