# Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$

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.

• 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. – M. Dus Apr 8 at 9:20
• 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 $R^n\cdot R^{-n}$. – YCor Apr 8 at 9:27