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

José Burillo

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain

E-mail Address: pep.burillo@upc.edu

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and

Victor Guba

Vologda State University, 6 S.Orlov Street, Vologda 160600 Russia

E-mail Address: guba@uni-vologda.ac.ru

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https://doi.org/10.1142/S0218196717500011Cited by:1 (Source: Crossref)

Abstract

Let $F (p)$ , $p \geq 2$ 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 $p$ .

Communicated by A. Olshanskii

Keywords:

AMSC: 20F32, 05C25

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