Nonlinearity of matrix groups
Abstract
Description
The aim of this note is to answer a question by Guoliang Yu of whether the group $EL_3(Z<x,y>)$, where $Z<x,y>$ is the free (non-commutative) ring, has any faithful linear representations over a field. We prove, in particular, that for every (unitary associative) ring $R$, the group $EL_3(R)$ has a faithful finite dimensional complex representation if and only if $R$ has a finite index ideal that has a faithful finite dimensional complex representation.
8 pages, no figures; v3: a question about EL_2 from the first version is answered
8 pages, no figures; v3: a question about EL_2 from the first version is answered