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Bulletin of the London Mathematical Society Advance Access originally published online on May 6, 2008
Bulletin of the London Mathematical Society 2008 40(4):593-603; doi:10.1112/blms/bdn032
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© 2008 London Mathematical Society

Some power of an element in a Garside group is conjugate to a periodically geodesic element

Eon-Kyung Lee

Department of Applied Mathematics
Sejong University
Seoul 143-747
Republic of Korea

Sang-Jin Lee

Department of Mathematics
Konkuk University
Seoul 143-701
Republic of Korea
sangjin@konkuk.ac.kr

Received 4 April 2006. Revision received 18 May 2007.

We show that for each element g of a Garside group, there exists a positive integer m such that gm is conjugate to a periodically geodesic element h, an element with |hn|D = |n| · |h|D for all integers n, where |g|D denotes the shortest word length of g with respect to the set D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside group, its stable super summit set.

2000 Mathematics Subject Classification 20F36, 20F10.


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