On groups acting freely on a tree |
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| Authors: | Ulrich Tipp |
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| Affiliation: | (1) Department of Fisheries and Wildlife, Oregon State University, Corvallis, OR 97331, USA;(2) Present address: Division of Forestry and Wildlife, State of Hawaii, 19 E. Kawili St., Hilo, Hawaii 96720, USA;(3) Institute of Pacific Islands Forestry, USDA Forest Service, 60 Nowelo St., PO Box 4370, Hilo, Hawaii 96720, USA |
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| Abstract: | Suppose we are given a group GmitGamma and a tree X on which GmitGamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers ad: = # {w ? vertX : w=gv, g ? G , d(v0,w)=d }a_d:= # {win {rm {vert}}X : w=gamma {v}, gamma in {mitGamma} , d({v}_0,w)=d } if d? ￥drightarrow infty , where v, vo are some fixed vertices in X.¶ In this paper we consider the case where GmitGamma is a finitely generated group acting freely on a tree X. The growth function ?ad xdtextstylesumlimits 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 GmitGamma 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. |
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