Simple closed curves, word length, and nilpotent quotients of free groups

Khalid Bou-Rabee and Asaf Hadari

Vol. 254 (2011), No. 1, 67–72

DOI: 10.2140/pjm.2011.254.67

Abstract

We consider the fundamental group π of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of π has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that shows that π has infinite diameter with respect to this set. We also give a general criterion for a finitely generated group equipped with a generating set to have this property.

Keywords

Simple closed curves, word length, nilpotent groups

Mathematical Subject Classification 2010

Primary: 57M05

Secondary: 20E26

Milestones

Received: 21 January 2011

Accepted: 5 April 2011

Published: 7 February 2012