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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Published electronically: 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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Additional Information

Olga Kharlampovich
Affiliation: Department of Mathematics and Statistics, Hunter College CUNY, 695 Park Avenue, New York, New York 10065

Alexei Myasnikov
Affiliation: Department of Mathematical Sciences, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, New Jersey 07030

Vladimir Remeslennikov
Affiliation: Department of Mathematics, Omsk State University, 55-A Prospect Mira, Omsk, Russia 644077

Denis Serbin
Affiliation: Department of Mathematical Sciences, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, New Jersey 07030

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05376-1
PII: S 0002-9947(2012)05376-1
Received by editor(s): August 9, 2009
Received by editor(s) in revised form: March 22, 2010, and May 3, 2010
Published electronically: January 31, 2012
Article copyright: © Copyright 2012 American Mathematical Society