Nielsen theory on infra-nilmanifolds modeled on the group of uni-triangular matrices

Abstract

Let Nilm be the group of m×m uni-triangular matrices. Then it is a connected and simply connected (m− 1)-step nilpotent Lie group. Using the averaging formulas, we compute the spectra of the Lefschetz, Nielsen and Reidemeister (coincidence) numbers of maps on infra-nilmanifolds modeled on Nilm. As a byproduct, we prove that the Bieberbach groups of Nilm (m ≥ 4) with Γm as its nil-radical satisfy the R∞ property.