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Boston College

HMI

# Pro-p groups and low dimensional topology

## Schedule of Talks

 Time Speaker Talk Thursday 9.00 - 10.00 Marc Lackenby Links between low-dimensional topology and profinite group theory 10:15 - 11:15 Alexander Lubotzky Property tau and Hyperbolic Manifolds 11:15 - 11:45 DISCUSSION 11:45 - 1:30 LUNCH 1:30 - 2:30 Andrzej Zuk Automata Groups 2:45 - 3:45 Peter Shalen 4-free groups and geometry of hyperbolic manifolds 4:00 - 5:00 Martin Kassabov Finite simple groups and expander graphs Friday 9:00 - 10:00 Dan Segal Words 10:15 - 11:15 Alan Reid Heegaard genus and Property tau for hyperbolic 3-manifolds 11:15 - 11:45 DISCUSSION 11:45 - 1:30 LUNCH 1:30 - 2:30 Tim Cochran Homology and p-series of Groups 2:45 - 3:45 Andrei Jaikin-Zapirain On  the p-gradient of finitely presented groups 4:00 - 5:00 Shelly Harvey Iterated torsion-free abelian covers and L^2-Betti numbers of 3-manifolds Saturday 9:00 - 10:00 Laurent Bartholdi Amenability of Groups and Algebras 10:15 - 11:15 Nigel Boston Random Pro-p Groups and Random Galois Groups 11.15 - 12.15 DISCUSSION END OF WORKSHOP

## Format

The three day directed workshop  will consist of background talks in the morning, followed by a question-and-answer session, and then afternoon lectures which will describe current research.

The first two days of the workshop there will be two morning talks and three afternoon talks. On the last day, there will be two morning talks followed by a discussion session in which participants will be encouraged to describe parallels that they observed between the problems and techniques used in the various fields. A list of observed parallels, open problems, and likely future research directions will be compiled for general dissemination.

The HMI will publish  the workshop lectures and results as a series of notes. The notes will be disseminated in paper form to other institutes as well as be posted on the HMI website.

## Abstracts

### Marc Lackenby (Oxford University)

Links between low-dimensional topology and profinite group theory

Abstract: I'll give a survey of some of the known links between these two areas. On the one hand, most of the key open questions in 3-manifold theory are concerned with finite-sheeted covering spaces, and so profinite group theory has obvious applications here. On the other hand, topological and geometric methods seem to be able to establish facts about the finite index subgroups of finitely presented groups that are not obvious algebraically. I'll also suggest some possible areas of future research, particularly in relation to groups with fast subgroup growth.

### Alexander Lubotzky (Hebrew University)

Property tau and Hyperbolic Manifolds

We will give a short introduction to Property T, property tau and expanders. Then we will show how these concepts were used to solve some cases of the 'virtual first Betty number conjecture' for hyperbolic manifolds. We will also give some introduction to the recent idea of Lackenby how to use these concepts to try to prove the virtual Haken conjecture for hyperbolic 3-manifolds. If time permits we will connect it also with pro-p groups.

Automata Groups

### Peter Shalen (University of Illinois, Chicago)

4-free groups and geometry of hyperbolic manifolds
This is joint work with Marc Culler.

A group Gamma is said to be k-free, for a given positive integer k, if every subgroup of Gamma with rank at most k is free. We prove that if M is a closed, orientable hyperbolic 3-manifold such that pi_1(M) is 4-free, then M contains a point P which is (log 7)-semithick. This means that there is a cyclic subgroup Z of pi_1(M,P) such that every homotopically non-trivial loop based at P which has length < log 7 represents an element of Z.

I will discuss the proof of this result and some applications to the study of hyperbolic volume. The proof involves a deep argument about the nerve of a covering of hyperbolic 3-space by displacement cylinders, and uses a recent group-theoretic result, recently proved by Kent and independently by Louder-McReynolds, that if two rank-2 subgroups of a free group have a proof, using Louder's folding machinery on graphs of spaces, have rank--2 intersection then they have rank-2 join. The applications to volume involve finding a (log 7)-semithick point of M and a (log 3)-thick point of M which are separated by a suitably large distance. Both the proof of the theorem and the new arguments in the applications make strong use of the generalized log(2k-1) theorem.

### Martin Kassabov (Cornell University)

Finite simple groups and expander graphs
joint work with A.Lubotzky and N. Nikolov

A finite graphs with large spectral gap are called expanders. These graphs have many nice properties and have many applications. It is easy to see that a random graph is an expander but constructing an explicit examples is very difficult. All known explicit constructions are based on the group theory --- if an infinite group G has property T (or its variants) then the Cayley graphs of its finite quotients form an expander family.

This leads to the following question: For which infinite families of groups G_i, it is possible to find generating sets S_i which makes the Cayley graphs expanders?

The answer of the question is known only in few cases. It seems that if G_i are far enough from being abelian then the answer is YES. However if one takes `standard' generating sets the resulting Cayley graphs are not expanders (in many cases).

I will describe a recent construction which answers the above  question in the case of the family of almost all finite simple groups. If S is a FSG it is possible to construct explicit generating sets F_S, such that the Cayley graphs C(S,F_S) are expanders, and the expanding constant can be estimated.

### Dan Segal (Oxford University)

Words

A survey of recent and older results about verbal width of  groups, with an emphasis on profinite groups.

### Alan Reid (University of Texas, Austin)

Heegaard genus and Property tau for hyperbolic 3-manifolds

We will discuss the proof and applications of the following result: Every finitely generated Kleinian group has a cofinal tower of finite index normal subgroups that has Property tau.

### Tim Cochran (Rice University)

Homology and p-series of Groups

We discuss conditions under which a group homomorphism induces  isomorphisms (respectively) on quotients of the group by the terms of  the lower central p-series, the derived p-series, or on the pro-p  completions. These homological conditions generalize Stallings' theorem  that a Z_p homology equivalence induces an isomorphism on the pro-p  completions. We give some applications to 3-manifolds. We give examples  of hyperbolic rational homology 3-spheres with the same pro-p completion  as a 3-manifold with nilpotent fundamental group. We give examples of  hyperbolic rational homology 3-spheres that have the same pro-p  completion as Z_p*Z_p...*Z_p, but admit no map onto this free product.

### Shelly Harvey (Rice University)

Iterated torsion-free abelian covers and L^2-Betti numbers of  3-manifolds

We will investigate L^2-Betti numbers associated to certain  iterated torsion-free abelian covers of 3-manifolds. In the case that the group of the cover is residually finite, these can be interpreted as the (weighted) limit of the betti numbers of some finite covers. We will show that the L^2-Betti numbers satisfy a certain monotonicity relation.  We will also give some  examples of 3-manifolds where the first Betti numbers cannot grow linearly in certain types of finite covers.

### Laurent Bartholdi (Ecole Polytechnique Federale Lausanne)

Amenability of Groups and Algebras

The amenable/nonamenable dichotomy is an important tool in Group Theory, in part because of its various algebraic, geometric and analytic interpretations. Another important invariant is "word growth"; if a group has subexponential growth then it is amenable, while the converse does not hold -- but see below!

I will define a natural notion of amenability for associative algebras (first considered by Gromov), and explain its relation to amenability of groups.

I will then prove the following result: let $G$ be a group, and let $(\gamma_n)$ be its lower central series. If $G$ is amenable, then the rank of $\gamma_n/\gamma_{n+1}$ increases subexponentially in $n$. This answers a question by Vershik; the proof resides in relating amenability to growth in an appropriate algebra.

As a corollary, all the groups constructed by Golod and Shafarevich groups are non-amenable. This gives the first examples ofnon-amenable, residually-finite torsion groups.

### Nigel Boston (University of Wisconsin, Madison)

Random Pro-p Groups and Random Galois Groups

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 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.

 Trinity College Dublin, College Green, Dublin 2. Tel: +353-1-608-1000. |