Limit groups with respect to Thompson's group F and other finitely generated groups

Abstract

Let F be the (Thompson's) group < x_0, x_1 | [x_0x_1^-1, x_0^-ix_1 x_0^i], i=1,2 >. We study the structure of F-limit groups. Let G_n= < y_1,..., y_m, x_0,x_1 | [x_0x_1^-1,x_0^-1x_1x_0],[x_0x_1^-1,x_0^-2x_1x_0^2], y_j^-1g_j,n(x_0,x_1), 0<j<m+1 >, where g_j,n(x_0,x_1) belongs to F, be a family of groups marked by m+2 elements. If the sequence (G_n)_n<w is convergent in the space of marked groups and G is the corresponding limit we say that G is an F-limit group. Primarily the paper is devoted to the study of F-limit groups. The results are based on some theorems concerning laws with parameters in F. In particular several constructions of such laws are given. On the other hand we formulate some very general conditions on words with parameters w(y,a_1,...,a_n) over F which guarantee that the inequality w(y,a') is not equal to 1 has a solution in F. Some of the results are of a more general nature and can be applied to study limit groups with respect to other finitely generated groups and classes of finitely generated groups, in particular to the case of the Grigorchuk group.