## 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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### References:

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