On Saturday, April 7, UW-Madison will host a one-day symposium on pro-p groups, their completed group algebras, and applications of these to number theory. All visitors are welcome, and funding may be available; please contact Jordan Ellenberg if you're interested in coming. Funding for this conference was provided by NSF-CAREER grant DMS-0448750 and by the Sloan Foundation.
All talks will be held in Van Vleck Hall, room 901.
Title: "Ideals in Iwasawa algebras"
Abstract: Michael Harris [J. Algebra, 1980] claimed that whenever H is a sufficiently large closed subgroup of a compact p-adic analytic group G, the induced module Zp[[G/H]] for the Iwasawa algebra Zp[[G]] of G has a nonzero annihilator. In 2003 Jordan Ellenberg found a gap in his paper, and since then it has been an open question as to whether or not there are any "interesting" nonzero two-sided ideals in these Iwasawa algebras, where we regard an ideal as being uninteresting if it is closely related to a closed normal subgroup of G. I will talk about some of my very recent work, joint with James Zhang and Feng Wei, which settles this question in the case when G is an open subgroup of SL_2(Zp).
Title: "Golod-Shafarevich groups in topology and number theory"
Abstract: Informally speaking, a finitely generated group is called Golod-Shafarevich if it has a presentation with a ``small'' set of relators (the definition makes sense both for discrete and pro-$p$ groups). In 1964, Golod and Shafarevich proved that groups satisfying such condition are necessarily infinite and used this criterion to solve two outstanding problems: the construction of infinite finitely generated periodic groups and the construction of infinite Hilbert class field towers.
An important class of discrete Golod-Shafarevich groups consists of the fundamental groups of compact hyperbolic 3-manifolds or, equivalently, cocompact torsion-free lattices in $SO(3,1)$. In 1983, Lubotzky used this fact to prove that arithmetic lattices in $SO(3,1)$ do not have the congruence subgroup property. More recently, Lubotzky and Zelmanov proposed a group-theoretic approach (based on Golod-Shafarevich techniques) to an even more ambitious problem, Thurston's virtual positive Betti number conjecture. This approach led to the following question: is it true that Golod-Shafarevich groups never have property $(\tau)$? I will show that the answer to the above question is negative in general and briefly describe examples of Golod-Shafarevich groups with property $(\tau)$ (in fact, property $(T)$) which are given by lattices in certain topological Kac-Moody groups over finite fields.
I will finish by comparing 3-manifold groups with the well-known number-theoretic examples of Golod-Shafarevich pro-$p$ groups given by the Galois groups of maximal pro-$p$ extensions of $\mathbb Q$ unramified outside of a finite set of primes.
Title: "Random pro-p groups and random Galois groups"
Abstract: Dunfield and Thurston studied how the distribution of finite quotients of a random g-generator g-relator abstract group compares with that of the fundamental group of a random 3-manifold obtained from a genus-g Heegard splitting. We consider the analogous questions for random g-generator g-relator pro-p groups and for Galois groups of maximal pro-p extensions unramified away from a finite set S of primes with |S| = g.
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