Garside groups are strongly translation discrete

In this paper we show that all Garside groups are strongly translation discrete, that is, the translation numbers of non-torsion elements are strictly positive and for any real number r there are only finitely many conjugacy classes of elements whose translation numbers are less than or equal to r. It is a consequence of the inequality "inf(s)(g) <= inf(s)(g(n))/n < inf(s) (g) + 1" for a positive integer it and an element g of a Garside group G where inf(s) denotes the maximal infimum for the conjugacy class. We prove the inequality by studying the semidirect product G(n) = Z proportional to G(n) of the infinite cyclic group Z and the cartesian product G(n) of a Garside group G, which turns out to be a Garside group. We also show that the root problem in a Garside group G can be reduced to a conjugacy problem in G(n), hence the root problem is solvable for Garside groups. (c) 2006 Elsevier Inc. All rights reserved.