On the strong algebraic eigenvalue conjecture for sofic groups.
11:30, Aula 520, UAM
Let $G$ be a countable group and let $K$ be a subfield of the field of
complex numbers $CC$. Let $a$ be an element in the group algebra
$K[G]$. Consider the associated operator $phi_a:l^2(G)to l^2(G)$
that acts as the right multiplication by $a$ (or, for the analysts,
as the right convolution): $phi_a: vmapsto va$. We say that
$lambdain CC$ is an eigenvalue of $a$ if there exists $0 e vin l^2(G)$ such that $phi_a(v)=lambda v$.
If $G$ is finite, then it is clear that $lambda$ is algebraic over
$K$. The strong algebraic eigenvalue conjecture claims that this
holds for an arbitrary group $G$. This conjecture was formulated by
Dodziuk, Linnell, Mathai, Schick, and Yates in 2003 in the case
when $K$ is the field of algebraic numbers.
In my talk I will explain the ideas of the proof of the conjecture for
arbitrary $K$ when $G$ is a sofic group.