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 “infs(g) 6 infs(g n) n < infs(g) + 1” 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 G(n) = Z⋉G of the infinite cyclic group Z and the cartesian product G 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.