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ISSN 1088-6850(e) ISSN 0002-9947(p)

Groups with free regular length functions in $\mathbb{Z}^n$

Authors: Olga Kharlampovich, Alexei Myasnikov, Vladimir Remeslennikov and Denis Serbin
Journal: Trans. Amer. Math. Soc. 364 (2012), 2847-2882
MSC (2010): Primary 20E08, 20F65
Posted: January 31, 2012
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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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