South England Profinite Groups Meetings

Joint Research Group funded by LMS Scheme 3
Academic Year 2012-13

This page keeps record of the meetings held during the year 2012-13.
Please click here to move to events scheduled for this year.

  • Background: Profinite and asymptotic methods have been successfully applied to general problems of group theory. In addition there are interesting links to number theory, geometry and other areas.
  • Activities: During the academic year 2012-13 we are holding a series of three meetings in order to
    - provide a platform for young mathematicians from across the south of England to meet and discuss their research in profinite groups, asymptotic group theory and related topics,
    - get graduate students and postdoctoral fellows involved by giving them the opportunity to discuss and possibly present their individual work,
    - give the participants an opportunity to widen their research horizons and their knowledge base by familiarising themselves with research adjacent to their own,
    - foster and develop collaborative links both within the group and with selected experts from the UK and abroad.
  • Participants: We welcome and encourage interested mathematicians at all levels to join us in our meetings.
    The Joint Research Group receives financial support from the London Mathematical Society and has therefore limited funds to reimburse travel expenses of UK-based students and young mathematicians. Please contact the organisers if you wish to apply for such reimbursements.
    For UK-based mathematicians with parental duties the LMS has a scheme which allows participants of meetings like ours to apply for a supplementary grant to help covering childcare costs; see here for further information.
  • Organisers: Corresponding organisers of the Joint Research Group are Benjamin Klopsch (Royal Holloway, University of London), Nikolay Nikolov (Univesity of Oxford) and Christopher Voll (University of Southampton).
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Calendar of Events

Past Meetings

Next Meeting

Cohomology in Profinite Groups

28 June 2013 at the Royal Holloway University of London

Introduction to Profinite Groups and Cohomology I and II: These talks will serve as introductory talks to both profinite groups and cohomology and in particular to cohomology in profinite groups.

Low Dimensional Group Theory: Some or all of the following topics will be discussed: free groups, the theory of ends, one relator groups, Poincare' duality groups of cohomological dimension 2, the surface group conjecture, groups of cohomological dimension 2, Poincare' duality groups of cohomological dimension 3

Abstract and Continuous Extensions of p-adic Lie Groups: In my talk I will discuss abstract versus topological extensions of compact p-adic Lie groups and applications to locally compact groups.  In particular I will report on joint work with Yiftach Barnea and Andrei Jaikin-Zapirain.  One of our results is the following.  Let H be an open subgroup H of a Chevalley group G(F) over a (non-arichimedean) local field F of characteristic zero.  Then every extension E of a finite group by H is topological.  Moreover, if H is compact then E is profinite.

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Complete Calendar of Events

One-Day Meeting: Lattices in Locally Compact Groups

One-Day Meeting: Analytic and Linear Pro-p Groups over Non-p-adic Pro-p Rings

One-Day Meeting: Cohomology in Profinite Groups

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Past Meetings

Lattices in Locally Compact Groups

14 Decmber 2012 at the University of Oxford

  • Description: Lattices provide lots of interesting concrete examples of infinite finitely presented groups which can have widely different
    properties: from being residually finite (in linear Lie groups) to infinite simple groups.
    This meeting aims to introduce some of the recent results and methods in this field to a non-expert audience.
  • Schedule:

    12-1pm Lunch in the Common Room

    1-2pm Cayley graphs of Fuchsian surface groups versus hyperbolic graphs, Caroline Series

    2:15-3:15pm Deformations and rigidity of lattices in soluble Lie groups, Benjamin Klopsch

    3:15-4pm coffee/tea

    4-5pm Some results and questions concerning lattices in totally disconnected groups, Tsachik Gelander

  • Speakers: Caroline Series (Warwick), Benjamin Klopsch (Otto-von-Guericke-Universität Magdeburg and Royal Holloway, University of London), Tsachik Gelander (Jerusalem)

  • Abstracts:

Cayley graphs of Fuchsian surface groups versus hyperbolic graphs: Most results about the Cayley graph of a hyperbolic surface group can be replicated in the context of more general hyperbolic groups. In this talk I will discuss two results about such Cayley graphs which I do not know how to replicate in the more general context.

Deformations and rigidity of lattices in soluble Lie groups: Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of Aut(G) on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev and Saitô for lattices in nilpotent Lie groups and in solvable Lie groups of real type. We prove that the deformation space of every Zariski-dense lattice Γ in G is finite and Hausdorff, provided that the maximal nilpotent normal subgroup of G is connected.  I will introduce all necessary notions and try to motivate and explain this result.

Some results and questions concerning lattices in totally disconnected groups: I'll discuss some results about lattices in totally disconnected locally compact groups, elaborating on the question: which classical results for lattices in Lie groups can be extended to general locally compact groups. For example, in contrast to Borel's theorem that every simple Lie group admits (many) uniform and non-uniform lattices, there are totally disconnected simple groups with no lattices. Another example concerns with the theorem of Mostow that lattices in connected solvable Lie groups are always uniform.

This theorem cannot be extended for general locally compact groups, but variants of it hold if one implants sufficient assumptions. At least 90% of what I intend to say is taken from a paper and an unpublished preprint written jointly with P.E. Caprace, U. Bader and S. Mozes. If time allows, I will also discuss some basic properties and questions regarding Invariant Random Subgroups.

  • Location: University of Oxford, Seminar room L3 in the Mathematical Institute 23-27 St Giles, Oxford, OX1 3LB.

  • Date: 14 December 2012 at 12-5pm

  • Local Organiser: Nikolay Nikolov

  • Support: The meeting is funded by the LMS Scheme 3 and the Oxford EPSRC Platform Grant.
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Analytic and Linear Pro-p Groups over Non-p-adic Analytic Pro-p Rings

22 March 2013 at the University of Cambridge

Analytic pro-p groups over pro-p domains - an introduction: The theory of Lie groups is highly developed in characteristic zero.
Over the last decades, the theory of compact p-adic analytic groups has led to profound applications in finite and infinite group theory.
In contrast to this, fairly little is known about analytic groups over local fields of positive characteristic p or, more generally, pro-p domains of characteristic p and higher Krull dimension.  I will briefly recall key aspects of the theory of p-adic analytic groups and touch upon applications of the available Lie theory to representation zeta functions.
Then I will focus on the structure theory of analytic groups over pro-p domains of positive characteristic.  Here I will outline both what we know and what we do not yet know.

On the representation and subgroup growth of F_p[[t]]-analytic profinite groups: I will give a brief introduction to some classical results about the subgroup and the representation growth functions of a profinite group and I will explain the main ideas behind few results on the subgroup and representation growth of some F_p[[t]]]-standard groups and some pro-p groups linear over local field or positive characteristic.

Applications of Pink's theorem to the study of linear pro-p groups over local fields of positive characteristic: R. Pink revolutionized our understanding of linear pro-p groups over local fields of positive characteristic by showing that they are close to open compact subgroup of algebraic groups. I will present two applications of Pink's theorem: (i) a non-ableian free pro-$p$ group is not linear over a local field and (ii) if G is an open compact subgroup of "nice enough" semisimple algebraic groups, then two random elements generate an open subgroup with probability 1. These are joint results with M. Larsen. Only superficial knowledge of algebraic groups is required for this talk.

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For past meetings during the academic year 2011-12 click here.

For past meetings during the academic year 2010-11 click here.

For past meetings during the academic year 2009-10 click here.

For past meetings during the academic year 2008-09 click here.

For past meetings during the academic year 2007-08 click here.

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last significant update 18-Nov-2013
please bring corrections to the attention of Yiftach Barnea