RESEARCH ARTICLE

Word length in symmetrized presentations of Thompson's group $F$

Matthew Horak^a, Alexis Johnson^b, Amelia Stonesifer<sup>c

a</sup> University of Wisconsin-Stout, Menomonie, WI 54751, USA
^b Grand Valley State University, Allendale, MI 49401-9403, USA
^c St. Olaf College, Northfield, Minnesota 55057, USA

Abstract: Thompson's groups $F, T$ and $Z$ were introduced by Richard Thompson in the 1960's in connection with questions in logic. They have since found applications in many areas of mathematics including algebra, logic and topology, and their metric properties with respect to standard generating sets have been studied heavily. In this paper, we introduce a new family of generating sets for $F$, which we denote as $Z_n$, establish a formula for the word metric with respect to $Z_1$ and prove that $F$ has dead ends of depth at least $2$ with respect to $Z_1$.

Keywords: Thompson's group $F$, dead ends, diagram group.

MSC: 20F65

Received: 05.12.2011
Revised: 07.09.2012

Language: English

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