Subgroups of \(\mathrm{GL}_2\) and trees. (Q403131)

The authors first characterize the subsets of the Bruhat-Tits tree of \(\mathrm{PGL}_2(K)\), where \(K\) a complete valued field, that are the sets of fixed points \(C(G)\) of a subgroup \(G\) of \(\mathrm{GL}_2(K)\). When \(G\) is irreducible, the \(C(G)\) are the ``strips'' in the tree. Then they evaluate the shape of the strip \(C(G)\) in terms of algebraic invariants of the group \(G\). Two applications to representation theory are given. One is about a new generalization of Ribet's lemma on extensions. The other is about the trace-convergence of a sequence of representations.