Towers of algebraic curves uniformized by discrete subgroups of $PGL_2(k_w)\times E$

Ihara, in his article “On congruence monodromy problems” showed that for a non-Archimedean local field $k_v$ one can associate to the discrete subgroups of $PSL_2(\mathbf R)\times PSL_2(k_v)$ of a certain type towers of algebraic curves on which $PSL_2(k_v)$ acts as a group of automorphisms. In the present article Ihara's results are carried over by means of Mumford's non-Archimedean uniformization to an analogous class of discrete subgroups of $PGL_2(k_w)\times E$, with $k_w$ a non-Archimedean field (of arbitrary characteristic), and $E$ a topological group whose compact open subgroups form a fundamental system of neighborhoods of $1$.
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