Growth of positive words and lower bounds of the growth rate for Thompson’s groups F(p)

Abstract

Let [Formula: see text], [Formula: see text] be the family of generalized Thompson’s groups. Here, [Formula: see text] is the famous Richard Thompson’s group usually denoted by [Formula: see text]. We find the growth rate of the monoid of positive words in [Formula: see text] and show that it does not exceed [Formula: see text]. Also, we describe new normal forms for elements of [Formula: see text] and, using these forms, we find a lower bound for the growth rate of [Formula: see text] in its natural generators. This lower bound asymptotically equals [Formula: see text] for large values of [Formula: see text].