Group Actions Seminar: Kalka, Reid

The next Group Actions Seminar will be on Tuesday 22 May at the University of Sydney.
Our speakers will be Arkadius Kalka (Queensland) at 12 noon and Colin Reid (Louvain) at
3pm.  

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Date: Tuesday 22 May 

Time: 12 noon 

Location: Access Grid Room, Room 829, Carslaw Building, University of Sydney 

Speaker: Arkadius Kalka (Queensland) 

Title: Conjugacy in braid groups with applications in non-commutative and
non-associative cryptography 

Abstract: 

We review braid and Garside groups and the history of conjugacy in this groups.  Then we
also consider the friends of the conjugacy problem like the subgroup (subCP), shifted
(ShCP), simultaneous conjugacy (SCP) and the double coset problem (DCP).  In particular
we improved invariants for the SCP, developed the first deterministic algorithms for
ShCP, and for subCP and DCP for parabolic subgroups of braid groups.  This is based on
joint work with several coauthors from Bar-Ilan University, Ramat Gan, Israel.  

Further motivation for these problems comes from non-commutative public key
cryptography, and we discuss basic key agreement protocols.  Dehornoy’s shifted
conjugacy leads us to left-selfdistributive (LD) systems, multi-LD systems, and our new
idea of non-associative cryptography.  

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Date: Tuesday 22 May 

Time: 3pm 

Location: Access Grid Room, Room 829, Carslaw Building, University of Sydney 

Speaker: Colin Reid (Université catholique de Louvain) 

Title: Local Sylow theory of totally disconnected, locally compact groups 

Abstract: 

Totally disconnected, locally compact (t.d.l.c.)  groups are a class of topological
groups that occur naturally as automorphism groups of locally finite combinatorial
structures, such as graphs or simplicial complexes.  Compact totally disconnected groups
are known as profinite groups, which can also be characterised as inverse limits of
finite groups, and some familiar concepts from finite group theory generalise directly
to profinite groups.  The inverse limits of finite p-groups are known as pro-p groups,
and for these we have a generalisation of Sylow’s theorem: given a profinite group G,
every pro-p subgroup of G is contained in a maximal pro-p subgroup (a ’p-Sylow
subgroup’), and all p-Sylow subgroups of G are conjugate.  At the same time, profinite
groups play a key role in the general theory of t.d.l.c.  groups, because every
t.d.l.c.  group has an open profinite subgroup, and all such subgroups are
commensurable.  Thus we can develop a ’local Sylow theory’ for t.d.l.c.  groups, based
on the Sylow subgroups of their open compact subgroups.  Starting from an arbitrary
t.d.l.c.  group G, we produce a new t.d.l.c.  group, the ’p-localisation’ of G: this is
naturally determined by G up to isomorphism, embeds in G with dense image, and has an
open pro-p subgroup corresponding to a local Sylow subgroup of G.  I will describe the
construction and some properties of the p-localisation, illustrating the concepts with
the example of the automorphism group of a regular tree of finite degree.  

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