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Register a new userPolynomial maps over $p$-adics and residual properties of mapping tori of group endomorphisms
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We continue our study of residual properties of mapping tori of free group endomorphisms. In this paper, we prove that each of these groups are virtually residually (finite $p$)-groups for all but finitely many primes$p$. The method involves further studies of polynomial maps over finite fields and $p$-adic completions of number fields.
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The mapping torus of an endomorphism Phi of a group G is the HNN-extension G*_G with bonding maps the identity and Phi. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.
Lie groups over local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups. The text is both a research article and a worked out set of lecture notes for a mini-course held June 27-July 1, 2016 at the MATRIX research center in Creswick (Australia) as part of the Winter of Disconnectedness. The text can be read in parallel to the earlier lecture notes arXiv:0804.2234 which are devoted to automorphisms, with sketches of proof. Complementary aspects are emphasized.
We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=mathbf{G}(k_v). Let Gamma be an arithmetic lattice in G and let C=C(Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is hat{F}_{omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example Gamma=SL_2(mathcal{O}(S)), where mathcal{O}(S) is the ring of S-integers in k, with S={v}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Gamma on the Bruhat-Tits tree associated with G.
We give an algorithm which computes the fixed subgroup and the stable image for any endomorphism of the free group of rank two $F_2$, answering for $F_2$ a question posed by Stallings in 1984 and a question of Ventura.
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