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Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if there is a finite subset of $\mathbb {CP}^1$ which is invariant by every element of $R$.

For $n \geq 1$ set $R^n : = \{g_1\cdots g_n : g_i \in R\}$ (that is, the set of products of elements in $R$ of length $n$) and $R^n \cdot R^{-n}:=\{gh^{-1} : g,h \in R^n\}$. I'm looking for a reference for the following statement, which seems to be true:

Assume $R$ is non-elementary. Then there exists an $N \geq 1$ and an $M \geq1$, such that $R^{N}$ contains a loxodromic element and $R^{M} \cdot R^{-M}$ contains a non-elliptic element.

I have a somewhat tedious proof of this fact but I would guess that such a result is well-known (at least the one about $R^N$). A reference to a book or a paper containing these results or closely related ones would be of great help.

Thanks in advance for your answers.

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    $\begingroup$ So for the first question, you're asking if the semi-group generated by $R$ contains a loxodromic element. The answer is yes, it is given by Theorem 6.2.3 in the book by Das, Simmons and Urbanski : available on the arXiv at arxiv.org/pdf/1409.2155.pdf. I think that looking carefully at their proofs in Section 6 also gives the answer to your second question. $\endgroup$ – M. Dus Apr 8 at 9:20
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    $\begingroup$ The notation is a bit weird: from $n,R$ you define $R^n$, fine, and then $S^n$, while there's no $S$ involved... you could write it $R^n\cdot R^{-n}$. $\endgroup$ – YCor Apr 8 at 9:27

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