Profinite Properties of $\ell^2$-InvariantsGeometric Group Theory Oberseminar
by
Endenicher Allee 60/0-008
Mathezentrum
Although higher $\ell^ 2$ -Betti numbers $b^{ (2)}_{n}$
, $n \geq 2$ are in general not profi-
nite invariants, some interesting information on the $\ell^2$ -cohomology of
$S$-arithmetic groups still seems to be preserved under profinite com-
mensurability. This leads us to prove a stability result for another
`$\ell^2$ -invariant strictly related to the $\ell^2$-Betti numbers: the sign of the
Euler characteristic, which indeed can be seen as the alternating sum
of such $b^{2}_{n}$.
In this talk we will introduce the basics and motivations for being inter-
ested in Profinite rigidity and then we will discuss the key steps needed
to prove our result for S-arithmetic groups in terms of local-global
principles in the setting of Galois cohomology.
All of this is joint work with Holger Kammeyer.