Groups with free regular length functions in ℤⁿ

Kharlampovich, Olga; Myasnikov, Alexei; Remeslennikov, Vladimir; Serbin, Denis

doi:10.1090/S0002-9947-2012-05376-1

Groups with free regular length functions in $\mathbb {Z}^n$
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by Olga Kharlampovich, Alexei Myasnikov, Vladimir Remeslennikov and Denis Serbin PDF

Trans. Amer. Math. Soc. 364 (2012), 2847-2882 Request permission

Abstract:

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on $\mathbb {Z}^n$-trees give one a powerful tool to study groups. All finitely generated groups acting freely on $\mathbb {R}$-trees also act freely on some $\mathbb {Z}^n$-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for $\mathbb {Z}^n$-actions (which is not the case for $\mathbb {R}$-trees) comes along with a robust algorithmic theory. In this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on $\mathbb {Z}^n$-trees and give necessary and sufficient conditions for such actions.

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