Growth of Positive Words and Lower Bounds of the Growth Rate for Thompson’s Groups

Let F (p), p ≥ 2 be the family of generalized Thompson’s groups. Here F (2) is the famous Richard Thompson’s group usually denoted by F . We find the growth rate of the monoid of positive words in F (p) and show that it does not exceed p + 1/2. Also we describe new normal forms for elements of F (p) and, using these forms, we find a lower bound for the growth rate of F (p) in its natural generators. This lower bound asymptotically equals (p − 1/2) log 2 e + 1/2 for large values of p.