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 “ ” for a positive integer n and an element g of a Garside group G, where infs denotes the maximal infimum for the conjugacy class. We prove the inequality by studying the semidirect product of the infinite cyclic group and the cartesian product Gn 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.