p-adic Hurwitz groups

Herrlich showed that a Mumford curve of genus g > 1 over the p-adic complex field Cp has at most 48(g− 1), 24(g− 1), 30(g− 1) or 12(g− 1) automorphisms as p = 2, 3, 5 or p > 5. The Mumford curves attaining these bounds are uniformised by normal subgroups of finite index in certain p-adic triangle groups ∆p for p ≤ 5, or in a p-adic quadrangle group p for p > 5. The finite groups attaining these bounds are p-adic analogues of the Hurwitz groups arising from curves over C. We construct explicit infinite families of such groups as quotients of ∆p and p. These include groups of type PSL2(q) and PGL2(q), all arising as congruence quotients when p ≤ 5, and also various alternating and symmetric groups arising as noncongruence quotients of the groups ∆p. 2000 Mathematics Subject Classification 11F06, 20B25, 14G22, 20E06