Vol. 254, No. 1, 2011

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

Khalid Bou-Rabee and Asaf Hadari

Vol. 254 (2011), No. 1, 67–72

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.

Simple closed curves, word length, nilpotent groups
Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20E26
Received: 21 January 2011
Accepted: 5 April 2011
Published: 7 February 2012
Khalid Bou-Rabee
Department of Mathematics
University of Michigan
2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States
Asaf Hadari
Department of Mathematics
University of Chicago
5734 University Ave.
Chicago, IL 60637
United States