Seminario teoría de grupos

Andrei Jaikin 

(UAM)

On the strong algebraic  eigenvalue conjecture for sofic groups.

Jueves 8/2/2018


11:30, Aula 520, UAM



Abstract:
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.