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Given a Mumford curve X over Q p , we show that for every semistable model X of X and every closed point x of this semistable model, there exists a finiteétale cover Y of X such that every semistable model of Y over X has a vertical component above x. We then give applications of this to the tempered fundamental group. In particular, we prove that two punctured Tate curves Q p with isomorphic tempered fundamental groups are isomorphic over Qp.doi:10.4171/prims/122 fatcat:bvas37ux3nehhmqjk6pif7tmxi