- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:04:57
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2007
3
Conjugacy in Garside groups I: cyclings, powers and rigidity
Joan
Birman
Barnard College, Columbia University, NEW YORK, UNITED STATES
Volker
Gebhardt
University of Western Sydney, Penrith South DC NSW 1797, AUSTRALIA
Juan
González-Meneses
Universidad de Sevilla, SEVILLA, SPAIN
Garside groups, conjugacy problem, ultra summit set, rigidity, stable ultra summit set
In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where rigid means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X m is conjugate to a rigid element, then m can be bounded above by ||Δ||3. In the particular case of braid groups {Bn ; n ∈ ℕ}, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.
Group theory and generalizations
General
221
279
10.4171/GGD/12
http://www.ems-ph.org/doi/10.4171/GGD/12