Fuchsian groups and Eichler's result

Question

Let $G$ be a Fuchsian group of first kind contained in $\text{PSL}_2(\mathbb{R})$. A result of Eichler says, there exists a finite set $S\subset G$ such that any $\gamma$ in $G$ can be written as a product $\prod_{i=1}^{k} \gamma_i,$ where each $\gamma_i$ are either in $S,$ or power of some parabolic element coming from $S,$ for some $k\ll \log ||\gamma||.$ Howerver, the original Eichler's paper is in German, (http://matwbn.icm.edu.pl/ksiazki/aa/aa11/aa11111.pdf) and this is giving me trouble to understand the proof. Does anyone know if I can find the proof written in English somewhere else ?

Answer 1

This follows from Theorem 2(i) and Theorem 4 in The structure of words in discrete subgroups of $\mathrm{SL}(2,\mathbb{C})$, by Beardon.

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Since it isn't explicitly stated, I will roughly summarize/explain how you get the result.

Lets $D$ be a convex fundamental polygon for $G$, $S^*$ the associated generating set, and let $S$ be the set $S^*$ with the natural parabolics added (generators of maximal parabolic subgroup at each ideal vertex). For example, the natural generating set for $\mathrm{PSL}_2(\mathbb Z)$ will have $S^*=S$ since the parabolic is already there (guessing Eichler was inspired by Euclidean algorithm). If you have the "usual domain" for a Fuchsian group corresponding to a complete, once punctured, finite volume, hyperbolic torus, $S^*$ will be $\{A,B,A^{-1},B^{-1}\}$. The fundamental domain has four parabolic vertices, which get identified, so adding $P=ABA^{-1}B^{-1}$, its cyclic permutations, and inverses will give $S$. In this torus example consider the parabolic vertex corresponding to $P$, v, and note that

$$ D, AD, (AB)D, (ABA^{-1})D, (ABA^{-1}B^{-1})D=PD $$

all contain $v$. More generally you have that $v \in (P^k W) D$ where $W$ is an initial segment of $P$.

Beardon defines decomposition of elements in Fuchsian groups into chunks $C_i$ which split into two types: type I are elements which are not longer than some constant $m$ and type II are elements longer than $m$. Theorem 3 tells you that type II $C_i$ are basically parabolics in the sense that there is a parabolic vertex $v \in D$ such that

$$ v \in D, A_1D,\dots,(A_1 \cdots A_n )D=C_i D$$

where $A_i \in S^*$. That means that $C_i= (P^k W)D$ where $P$ is the parabolic at $v$ and $W$ is some initial segment of $P$.

Theorem 2 tells you that there are log many $C_i$ compared to norm and Theorem 4 gives the bounds if you "collapse" the type II pieces/split into parabolic part.