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Attainability of the minimal exponent of exponential growth for some Fuchsian groups. (English. Russian original) Zbl 1262.20037

Math. Notes 88, No. 1, 144-148 (2010); translation from Mat. Zametki 88, No. 1, 152-156 (2010).
From the introduction: Suppose that a group \(G\) is generated by a finite set of elements \(S\). By the growth function \(F_{G,S}(m)\) of the group \(G\) with respect to \(S\) we mean the function whose value is the number of elements of \(G\) expressible by group words of length not exceeding \(m\) in the alphabet \(S\). By the obvious inequality \(F_{G,S}(k+m)\leq F_{G,S}(k)F_{G,S}(m)\), the following limit exists: \(\lambda_{G,S}=\lim_{m\to\infty}(F_{G,S}(m))^{1/m}\). This limit is referred to as the exponent of exponential growth or the growth exponent of \(G\) with respect to the set of generators \(S\). Since only the growth exponents of this kind are treated in what follows, we omit below the word “exponential” for brevity. By the minimal growth exponent of \(G\) we mean the greatest lower bound \(\lambda_G=\inf_S\lambda_{G,S}\) over all finite sets of generators \(S\) of the group \(G\).
In this note, we prove that the minimal growth exponent is attained for the groups \[ H_{g,n}=\langle s_1,t_1,\dots,s_g,t_g\mid ([s_1,t_1]\cdots[s_g,t_g])^n=1\rangle\quad\text{with }n>1. \] These groups are naturally related to the fundamental groups of two-dimensional orientable surfaces and are also known as Fuchsian groups of signature \((g;n)\). Indeed, adding one generating element \(z=([s_1,t_1]\cdots[s_g,t_g])^{-1}\), one can represent the above presentation of the group \(H_{g,n}\) as the following canonical presentation of a Fuchsian group with one elliptic element \(z\) of order \(n\): \(\langle s_1,t_1,\dots,s_g,t_g,z\mid z[s_1,t_1]\cdots[s_g,t_g]=1,\;z^n=1\rangle\).
Theorem 1. The minimal growth exponent of the group \(G=H_{g,n}\) with \(n>1\) is attained at some set of generators which consists of \(2g\) elements.

MSC:

20F05 Generators, relations, and presentations of groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20F69 Asymptotic properties of groups
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