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 ?
1 Answer
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.
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.

$\begingroup$ Hi Paul, thanks for mentioning the reference. Beardon is first considering the Dirichlet polygon as a fundamental domain and then using the elements of side pairing elements to generate words. However, its not clear to me what type of elements they are. Can we say they are finite up to multiplication by some parabolic ? $\endgroup$– dragoboyCommented Sep 2, 2020 at 16:53

$\begingroup$ so you mean any element of $G*$ is either hyperbolic or a parabolic ? no elliptic element is there or I am missing something ? $\endgroup$– dragoboyCommented Sep 3, 2020 at 15:35

$\begingroup$ @dragoboy I am saying you can have any of the types. the PSL2(Z) case has an elliptic in $G^*$ (the order two element) and a parabolic. $\endgroup$– user35370Commented Sep 3, 2020 at 15:37

1$\begingroup$ @dragoboy The parabolic will fix $v$. The $W$ part won't fix $v$, it moves one of the other parabolic vertices of $D$ to $v$. $\endgroup$– user35370Commented Sep 5, 2020 at 23:34

1$\begingroup$ @dragoboy yes, exactly $\endgroup$– user35370Commented Sep 6, 2020 at 15:01