Regular geodesic languages for 2-step nilpotent groups

Summary

Introduction

One possibility to show that a group G with finite semigroup generating system S has rational growth series is to exhibit a regular language L ⊆ S^* consisting of geodesic words and mapped bijectively onto G. Machi and Schupp have even conjectured that the existence of such an L is equivalent to the rationality of the growth series ([2], conjecture 8.7).

The aim of this paper is to show that this approach does not work for nilpotent groups. More precisely, we show that if G is 2-step nilpotent with maximal free abelian quotient G and S is any finite set of semigroup generators for G, then for every regular and geodesic (with respect to G) language L ⊆ S^*, the natural map L → G has finite fibers. We conjecture that this holds for all nilpotent groups. If G is not virtually abelian, this implies in particular that L cannot be mapped bijectively onto G.

This also gives a counterexample to the conjecture of Machi and Schupp, for it is known that the discrete Heisenberg group with its standard generating set,

has rational growth series [3], but our theorem implies that no regular and geodesic language can be mapped bijectively onto H.

Notations and Definitions

Let G denote an arbitrary finitely generated group.