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I'm not sure if you're interested in classical groups over Q_p or over Z_p; in the latter case, one can often show that a closed subgroup H of G is in fact the whole group G once you know it projects …

I agree with the answers above. Allen Broughton at Rose-Hulman is a guy who has written a lot about automorphisms of Riemann surfaces: his paper Classifying finite group actions on surfaces of low g …

To help with searching: the set you ask about is usually called the "representation variety" or the "character variety." The study of representation varieties of surface groups (of which free groups …

My first stab at this would be to think of SL_2(Z,L) acting on the upper half plane. You can see what the cusps are and what conjugacy classes in SL_2(Z,L) they correspond to; mod out by the subgroup …

In a forthcoming paper with Venkatesh and Westerland, we require the following funny definition. Let G be a finite group and c a conjugacy class in G. We say the pair (G,c) is nonsplitting if, for e …

This fact is used in a nice way by Dunfield and Thurston to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (T …

The Frattini quotient G/Phi(G) is the maximal elementary abelian quotient of G; the group of automorphisms of G which act trivially on G/Phi(G) is a p-group, and so Aut(G) is solvable if and only if i …

See the theorem of Mann quoted by Jaikin in this question for a nice criterion guaranteeing topological finite generation in terms of the number of open subgroups of index N. See Pete Clark's answer …

Your question about the median is equivalent to: if the diamaeter of a group in some set of generators is d, how long does it take to generate half the group? The answer, as you've observed, depends …

I don't think Critch's reply above answers Harald's question; it seems to presume that the map F_2 -> SL_n(Z/pZ) factors through a chosen inclusion of SL_2(Z/pZ), while Harald wants pairs of elements …

By the way, I think that under your hypotheses, your question is really about group theory, not about algebraic geometry. Namely: the action of Galois on E[p^infty] x E'[p^infty] gives you a homomor …

I'll add a brief commment to Arne Semeets's thorough and useful answer. If I fix three rational conjugacy classes c_0, c_1, c_infty in a finite group G, then there are finitely many isomorphism class …

Almost but not quite. A congruence subgroup has to contain some $\Gamma(N)$. So if it is normal its image in $\text{SL}_2(\mathbb{Z})/\Gamma(N)$ is a normal subgroup of $\text{SL}_2(\mathbb{Z}/N\mat …

Re your question 3: Yes, for a given n there are only finitely many degree-n extensions of Q unramified outside S. Such an extension has discriminant bounded in terms of n and S and the number of de …

Here's a more general question which should give you a hint how to solve your problem: prove that the given assertion is true whenever the finite group Gal(L/K) has the property that every involution …