In this article we develop a theory of the so-called Mumford orbifolds, viz. rigid-analytic orbifolds covered by Mumford curves. General recipe for treating such orbifolds is given. The main result states a necessary and sufficient condition for abstract graphs of groups to be realized as discrete groups for Mumford orbifolds, which is useful for construct- ing several interesting non-Archimedean discrete groups, such as p-adic triangle groups.
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