The growth of torus link groups

2014/01/15 by Yoshiyuki Nakagawa, Nakagawa, Yoshiyuki, Makoto Tamura +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Geometric and Algebraic Topology #math.GR #math.GT #msc:20E06 #msc:20F05 #msc:57M25 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1401.3403

7 pages

arxiv created 2014/01/15 · arxiv updated 2014/01/16

Abstract

Let G be a finitely generated group with a finite generating set S. For g∈ G, let l_S(g) be the length of the shortest word over S representing g. The growth series of G with respect to S is the series A(t) = ∑_n=0^∞ a_n tⁿ, where a_n is the number of elements of G with l_S(g)=n. If A(t) can be expressed as a rational function of t, then G is said to have a rational growth function. We calculate explicitly the rational growth functions of (p,q)-torus link groups for any p, q > 1. As an application, we show that their growth rates are Perron numbers.

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