SUBGROUP DISTORTIONS IN NILPOTENT GROUPS

Abstract

The explicit formula for the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group is obtained. In particular, we prove that a function f: N → R can be realized (up to equivalence) as the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group if and only if f ∼ n^r for some nonnegative r ∈ Q. Considering lattices in Lie groups, we establish the analogous results for finitely generated nilpotent groups.

ACKNOWLEDGMENTS

The research has been supported by the RFFR grant 99–01–00894 and by the INTAS through the Network project 99-1224 “Combinatorial and Geometric Theory of Groups and Semigroups and its Applications to Computer Science”. I am grateful to A. Ol'shanskii for his hospitality in Vanderbilt University, where the main part of the article has been written. Also I would like to express my gratitude to P. de la Harpe, C. Pittet and E. Rips for useful comments, corrections and explanations.