Groups in Galway 2016

20-21 May 2016
NUI Galway Logo

Groups in Galway has been running on an annual basis since 1978. The scope of the conference workshop covers all areas of group theory, applications, and related fields. All who are interested are invited to attend. There is no conference fee. The webpage of last year's Groups in Galway conference is located here.


Conference Photos 1 2

Poster Prize Jone Uria Albizuri Feyishayo Olukoya Daher Al-Baydli

List of Participants


All talks will be in AM150 in the Arts Millenium Building of NUI Galway. The conference dinner will be held at the Westwood Hotel. It is 25 euro for regular participants and 15 euro for postgraduates and early career researchers.

Move the cursor over a title for an abstract.

Friday 20 May

10.00-10.45 John Burns Discrete Tori in Weyl groups and their applications
Abstract: Motivated by a question from Statistical Mechanics, G. Pfeiffer and I recently classified the conjugacy classes of maximal order Abelian subgroups of Weyl groups (discrete tori). We develop analogies with their counterparts in compact Lie groups, such as Cartan’s theorem. We describe some applications to configurations of exceptional lines on Del Pezzo surfaces , quaternionic interpretations of orbits roots and weights under discrete maximal tori and the theory of calibrations.

10.45-11.15 Coffee/tea

11.15-12.00 Shane O'Rourke A combination theorem for affine tree-free groups
Abstract: Isometric actions on $\Lambda$-trees (where $\Lambda$ is an ordered abelian group) have been studied by several authors, including Morgan, Shalen, Chiswell, Bass, Kharlampovich, Miasnikov, Remeslennikov and Serbin. In particular Bass showed how isometric actions of (vertex) groups on $\Lambda_0$-trees can be combined to give an isometric action (on a $\Z\times\Lambda_0$-tree) of the fundamental group of an associated graph of groups, provided certain compatibility conditions are met. Notably, the hyperbolic lengths of the embedded images $\alpha_e(g)$, $\alpha_{\bar{e}}(g)$ of elements $g$ of edge groups must match up. Groups that admit a free isometric action on a $\Lambda$-tree are a generalisation of free groups in that free groups correspond precisely to the case $\Lambda=\mathbb{Z}$. Affine actions are actions by dilations: one requires $d(gx,gy)=a_g d(x,y)$ where $a_g$ is an order-preserving group automorphism of $\Lambda$. In this talk we will show how certain combinations of groups can be equipped with an affine action on a $\Lambda$-tree. That is, if a graph of groups is given where the vertex groups have affine actions on $\Lambda_0$-trees, the fundamental group admits an affine action on a $\Lambda$-tree where $\Lambda=\Z\times\Lambda_0$, provided certain compatibility conditions are satisfied. Focussing on the case of free actions, we show that a large class of one-relator HNN extensions of free groups admit free affine actions on $\Lambda$-trees. Such HNN extensions cannot typically act freely by isometries because of the requirement that $\alpha_e(g)$ and $\alpha_{\bar{e}}(g)$ have the same hyperbolic length. Using recent work by various authors, we also show that groups that admit a free affine action on a $\Z^n$-tree with no inverted line are locally quasiconvex and relatively hyperbolic with nilpotent parabolic subgroups; they therefore have solvable word, conjugacy and isomorphism problems.

12.15-1.00 Collin Bleak On detecting solubility for finitely generated subgroups of the group PL_o(I)
Abstract: We describe an algorithm which determines in finite time whether or not any given finite set of generators determines a soluble subgroup of the group PL_o(I) of piecewise-linear orientation-preserving homeomorphisms of the unit interval. The algorithm works across the broad class of ``computable’’ subgroups of Pl_o(I), and makes strong use of dynamical arguments. Joint with Tara Brough and Susan Hermiller.

1.00-2.30 Lunch

2.30-3.00 Poster session with coffee/tea

3.00-3.45 Bob Oliver Automorphisms and extensions of fusion systems
Abstract: Fix a prime $p$. A fusion system $\mathcal{F}$ is \emph{tamely realized} by a finite group $G$ if $\mathcal{F}\cong\mathcal{F}_S(G)$ (for $S\in\textup{Syl}_p(G)$) and the natural map from $\textup{Out}(G)$ to $\textup{Out}(\mathcal{L}_S^c(G))$ is split surjective. Here, $\mathcal{L}_S^c(G)$ is the centric linking system for $G$. The fusion system $\mathcal{F}$ is \emph{tame} if it is tamely realized by some finite group. Tameness plays an important role when studying extensions of fusion systems, and through that when using fusion systems to classify certain classes of finite groups. It is also interesting in its own right as a means of describing automorphisms of the fusion and linking systems for $G$ (and also of its $p$-completed classifying space) in terms of $\textup{Out}(G)$. The goal of this talk is to explain these connections, and also to describe recent progress in proving tameness for fusion systems of finite simple groups.

4.00-4.45 Ellen Henke Normal subsystems of fusion systems and partial normal subgroups of localities
Abstract: Saturated fusion systems are categories generalizing important features of fusion in finite groups. Many concepts in finite group theory have analogues in the language of fusion systems. In particular, normal subsystems of fusion systems are defined. Broto, Levi and Oliver introduced centric linking systems to be able to study classifying spaces of fusion systems. It was proved by Andrew Chermak that there is a unique centric linking system associated to each saturated fusion system. For his proof he introduced the concept of a locality which can be thought of as a ``partial group'' with a multivariable product only defined on certain words. A centric linking system corresponds to a centric linking locality. Since localities are so group like, there is a very natural notion of a partial normal subgroup of a locality. I will report on a joint project with Andrew Chermak where we prove that there is a one to one correspondence between the normal subsystems of a fusion system and the partial normal subgroups of an associated linking locality.

5.00-5.45 Mark Lawson Boolean full groups
Abstract: This talk is about how to build interesting groups from interesting inverse monoids and will be entirely introductory. Boolean inverse monoids form a variety of algebras that can be viewed as non-commutative generalizations of unital Boolean algebras. In particular, non-commutative Stone duality relates Boolean inverse monoids to the \'etale topological groupoids whose spaces of identities are compact Boolean spaces. There are analogies between Boolean inverse monoids and $C^{\ast}$-algebras of real rank zero. In particular, there are Boolean inverse monoid analogues of AF, Cuntz, and Cuntz-Krieger $C^{\ast}$-algebras as well as the $C^{\ast}$-algebras associated with aperiodic tilings. In this paper, we study the structure of the groups of units of {\em simple} Boolean inverse monoids; they are precisely the minimal full subgroups of the group of homeomorphisms of the Stone space of the Boolean algebra of idempotents of the Boolean inverse monoid. In addition, these groups contain as subgroups infinite analogues of both the symmetric and alternating groups as first demonstrated by Nekrashevych.

6.30 - 7.30 Reception (Dangan Suite, The Westwood Hotel)

7.30 - Conference dinner (The Westwood Hotel)

Saturday 21 May

9.00-09.45 Nadia Mazza On a pro-p group of upper triangular matrices
Abstract: In this talk, we will discuss a pro-p group G whose finite quotient groups give your "favourite" Sylow p-subgroups of GL(n,p^f), for all positive integers n,f and p odd. Elaborating from work by Weir in the 50s and recent results by Bier and Holubowski, we will dip into the subgroup structure of G. Time permitting, we will also discuss field extensions, a p-adic variant of G and Hausdorff dimensions of some closed subgroups.

10.00-10.45 Peter Symonds Endotrivial modules for infinite groups

10.45-11.15 Coffee/tea

11.15-12.00 Francesco de Giovanni The murdered cardinal: a countably recognizable crime
Abstract: The influence of countable subgroups on the structure of groups of high cardinality is investigated.

12.15-1.00 Said Sidki From the Alternating Groups to Orthogonal Groups over Laurent Polynomial Rings
Abstract: The topic of this lecture are the simple looking group presentations in two parameters \begin{eqnarray*} Y(m,n) &=& \end{eqnarray*}% introduced in 1982, generalizing Carmichael's presentation of $Alt\left( m+2\right) $ (for $n=3$). It was conjectured then that $Y(m,n)$ are presentations for finite groups for all $m,n$. When $n\geq 5$ odd, the conjecture stated further that they are presentations of orthogonal groups in characteristic $2$. We will review the status of the problem and new approaches in the realm of orthogonal groups over Laurent polynomial rings, obtained together with Justin McInroy and Sergey Shpectorov.

Lecture videos

(Unfortunately lectures of Collin Bleak and Peter Symonds were not properly recorded.)


If you want to participate in the conference, please send us an email to with subject line "gig2016 registration" indicating your name, affiliation and whether you want to join the conference dinner (indicating special dietary requirements if they apply).


We have limited funding for postgraduate students and early career researchers. If you want to be considered for support, please send us the registration email with you CV by 19 March.

Poster prize

There will be a poster session for students, PhD students, post-doctoral research fellows and other young researchers, and research expenses prizes will be awarded to the top ranked posters. If you are not based in Galway, you can send a pdf file of your poster to, and he will get your poster printed locally (free of charge for the participant), in order to avoid transport damage.


Galway can be reached by public transportation from Dublin Airport, Shannon Airport and Knock Airport (also refered to as Ireland West Airport). From Dublin airport there are direct shuttle buses to Galway operated by Citylink and GoBus. You can also take a train from Dublin city. Bus Eireann also runs buses from Dublin Airport and have recently set us direct runs from Shannon Airport to Galway. Knock Airport now has bus connections to Galway. From Great Britain, Galway can also be reached by Rail and Sail.

Directions to NUI Galway by road can be found here .

Galway is a small city and you can reach any destination in the city centre comfortably by walking. It takes about 15 minutes from Galway Coach, Bus or Rail Station to the university (Google maps direction).

NUI Galway has a number of pay-and-display parking places for visitors. Cars parked in other spaces on the NUI Galway campus and not displaying a valid parking permit will be clamped.


The following hotels and guest houses are convenient for the NUI Galway campus:

Please contact Ireland West for further information about accommodation near NUI Galway.

For further information, please keep an eye on this website which will be updated regularly, or contact the organisers Ted Hurley and Sejong Park.

Groups in Galway 2016 is generously supported by