Nonlinearity of Matrix Groups

The aim of this short note is to answer a question by Guoliang Yu of whether the group EL3(Z〈x, y〉), where Z〈x, y〉 is the free (non-commutative) ring, has any faithful finite dimensional linear representations over a field. Recall that for every (associative unitary) ring R the group ELn(R) is the subgroup of GLn(R) generated by all n × n-elementary matrices xij(r) = Id + reij (r ∈ R, 1 ≤ i 6= j ≤ n). Clearly, if R has a faithful finite dimensional linear representation over a field, then the group ELn(R) also has a faithful finite dimensional linear representation over the same field. The conclusion is true even if R has an ideal of finite index that has a faithful finite dimensional representation (see Theorem 1). The converse implication (which would imply the negative answer to G. Yu’s question) should have been known for many years, but we could not find it in the literature. There are many results about isomorphisms between various matrix groups over (mostly commutative) rings from the original results of Mal’cev [Ma] to results of O’Meara [OM] to Mostow rigidity results [Mo]. There are also many result about homomorphisms of one general matrix group into another. Churkin [Ch] proved that the wreath product Z ≀ Z embeds into a matrix group over a field K of characteristic 0 if and only if the transcendence degree of K over its prime subfield is at least n (a similar result is proved in the case of positive characteristic). Hence SLn(K) cannot embed into SLm(K ) if K,K ′ are fields of characteristic 0 and the transcendence degree of K is bigger than the transcendence degree of K . Much stronger non-embeddability results for general linear groups over fields follow from the main result of Borel-Tits [BT] (we are grateful to Yves de Cornulier for pointing out to this reference). See also surveys [JWW] and [HJW]. The main result of the note is the following: