# Ping pong with parabolic isometries on Gromov hyperbolic spaces

For a group $$G$$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $$(X,d)$$, it is well known that you can construct free subgroups of $$G$$ via the ping pong argument with (sufficiently large powers of) two hyperbolic elements with distinct fixed points in $$\partial X$$.

In this question I ask for a reference for the case of ping pong with two parabolic isometries. This is perhaps well known in the case of $$X=\mathbb{H}^2$$, but I am interested in general Gromov hyperbolic spaces. I think the following should be true, but ideally I would like to cite it. Does anyone know of such a reference?

Let $$G$$ be a group acting by isometries (not necessarily properly) on a Gromov hyperbolic geodesic metric space $$(X,d)$$. Fix a basepoint $$o\in X$$.

Suppose

1. that $$f,g\in G$$ both induce parabolic isometries on $$(X,d)$$,
2. the fixed points of $$f$$ and $$g$$ in $$\partial X$$ are distinct, and
3. for any $$B>0$$ there exists $$N>0$$ such that for all $$n>N$$ we have $$d(o,f^n o),d(o,g^n o) >B$$.

Then for sufficiently large $$M$$ we have that $$f^M$$ and $$g^M$$ generate a free group $$F$$ of rank two, and, any non-trivial $$h\in F$$ is either hyperbolic on $$(X,d)$$ or $$h$$ is conjugate (in $$F$$) to a power of $$f$$ or $$g$$.

A reference would be ideal and greatly appreciated. Thank you.

• It seems to me that (3) follows from (1), in an arbitrary proper Gromov-hyperbolic space (in a proper metric space $X$ with isometry $f$, $(d(o,f^n(o)))_n$ is either bounded or tends to infinity. – YCor May 4 at 10:51
• Not a real answer. Lemme 2.3 of the book of Coornaert, Delzant and Papadopoulos gives a criterion when the product of two non-hyperbolic isometries is hyperbolic. In you case this should prove that for sufficiently large $M$ the product $f^{\pm M}g^{\pm M}$ is hyperbolic. With a little bit of work you should be able to make this argument work for any freely reduced product in $f^M$ and $g^M$. – Richard Weidmann May 5 at 19:33