ELA, Volume 7, pp. 59-72, July 2000, abstract. 

On the group GL(2,R[x])

Valeryi Fayiziev

Suppose that G is an arbitrary group and S is its subset such that 
S^{-1}= S.  Let gr(S) be the subgroup of G generated by S.
Denote by l_S(g) the length of element g in gr(S) relative to the 
set S.  Let V  be a finite subset of a free group F of countable 
rank and let the verbal subgroup V(F) be a proper subgroup of F.  
For an arbitrary group G, denote by \overline V(G) the set of 
values in the group G of all the words from the set V.
The present paper establishes the infinity of the set
{l_S(g), g in V(G)}, where G= GL(2,R[x]), S= \overline V(G) \cup
{ \overline V(G) }^{-1} for an arbitrary field R.