Archiv der Mathematik

, Volume 72, Issue 4, pp 261–269 | Cite as

On groups acting freely on a tree

  • Ulrich Tipp


Suppose we are given a group \(\mit\Gamma \) and a tree X on which \(\mit\Gamma \) acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers \(a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \}\) if \(d\rightarrow \infty \), where v, v o are some fixed vertices in X.¶ In this paper we consider the case where \(\mit\Gamma \) is a finitely generated group acting freely on a tree X. The growth function \(\textstyle\sum\limits a_d x^d\) is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where \(\mit\Gamma \) is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.


Rational Function Asymptotic Behavior Growth Function 
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Copyright information

© Birkhäuser Verlag, Basel 1999

Authors and Affiliations

  • Ulrich Tipp
    • 1
  1. 1.FB 9 Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, GermanyDE

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