p-adic Schottky space and Outer space

A classical complex Schottky group of rank $g \ge 1$ is a Kleinian group with a fundamental domain that is the complement of $2g$ disjoint disks in the Riemann sphere. Every compact Riemann surface of genus $g$ can be uniformized by such a Schottky group. The construction of Schottky groups works in the same way over a $p$-adic field, and the corresponding quotient space is again (the analytification of) a projective curve. But only curves with a totally degenerate reduction, the so called Mumford curves, can be uniformized this way.

The conjugacy classes of $g$-tuples of Möbius transformations freely generating a Schottky group are the elements of Schottky space. It is a $(3g − 3)$-dimensional analytic manifold on which the outer automorphism group $\textrm{Out}(F_g)$ of the free group $F_g$ acts naturally.

On the other hand, Outer space $\textrm{CV}_g$, an (almost) simplicial complex of dimension $3g − 3$, classifies marked metric stable graphs of genus $g$ up to homotopy equivalence. Again, $\textrm{Out}(F_g)$ acts in a natural and discontinuous way on $\textrm{CV}_g$.

The link between the two spaces comes from the Bruhat-Tits tree of the $p$-adic field: a Schottky group acts on a simplicial subtree which carries a natural metric; the quotient graph determines a point in Outer space. This observation leads to an interesting map from $p$-adic Schottky space to Outer space.