Dead ends and rationality of complete growth series

TL;DR

This paper investigates the algebraic properties of complete growth series of certain groups, revealing obstructions like dead ends to their rationality and algebraicity, especially in Heisenberg and lamplighter groups.

Contribution

It establishes new obstructions to the rationality and algebraicity of complete growth series in specific groups, extending understanding of their algebraic structure.

Findings

Dead ends obstruct $ ext{NG}$-rationality.

Complete series of $H_3( ext{Z})$ are not $ ext{NG}$-algebraic.

Higher Heisenberg and lamplighter groups' series are not $ ext{NG}$-rational.

Abstract

The complete growth series of a finitely generated group is given by $\sum_{n\ge 0} A_ns^n$, where $A_n$ is the sum of elements of length $n$ in the group semiring. We study the $\mathbb NG$-rationality and $\mathbb NG$-algebraicity of such series. We show that having dead ends of arbitrarily large depths is an obstruction to $\mathbb NG$-rationality. In the case of the $3$-dimensional Heisenberg group $H_3(\mathbb Z)$, we prove that the complete series is not $\mathbb NG$-algebraic for any generating set. Dead ends are also used to show that complete growth series of higher Heisenberg groups are not $\mathbb NG$-rational for specific generating sets. Using a more general version of this obstruction, we prove that complete growth series of some lamplighter groups are not $\mathbb NG$-rational either.