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Groups in Galway 2017

18-20 May, 2017

Groups in Galway has been running on an annual basis since 1978. The scope of the conference covers all areas of group theory, applications, and related fields. ALL NUI Galway is committed to equality of opportunity for all staff, students and visitors irrespective of sex, marital status, family status, sexual orientation, religious belief, age, disability, race, colour, nationality or ethnic or national origin, membership of the travelling community or perceived political belief or affiliation. who are interested are invited to attend. There is no conference fee. The web page of last year's Groups in Galway conference is located here.

Speakers

Confirmed speakers at the moment include List of participants

Schedule

Below is a tentative schedule which may still change but only minimally. All talks will be in AM150 in the Arts Millenium BUilding.

Move the cursor over a title for a short abstract.

Thursday 18 May

14.00‒14.45 Conchita Martínez-Pérez Homological dimension of solvable goups We show that the homological dimension of any solvable group over any (commutative) coefficient ring, when finite, equals its Hirsh rank. We will also discuss the history of this problem which was believed to be completely solved for a while. This is a joint work with Peter Kropholler.

15.00‒15.45 Markus Szymik Homology of the automorphism groups of free nilpotent groups The automorphism groups of free nilpotent groups of various classes interpolate between the integral general linear groups and the automorphism groups of free groups. In this talk, the focus is on their homology and connections to functor categories.

15.45‒16.30 Coffee/tea

16.30‒17.15 Derek Holt A new method for verifying the hyperbolicity of finitely presented groups A finitely presented group is called hyperbolic if geodesic triangles in its Cayley graph are uniformly thin or, equivalently, if its Dehn function is linear.

The programs in the author's KBMAG package, which is implemented in Magma, can verify hyperbolicity of a give finitely presented group.

In this talk we describe new methods for proving hyperbolicity and for estimating the Dehn function that are based on small cancellation theory and the analysis of the curvature of van Kampen diagrams for the group. A first version of a Magma implementation is available.

These methods are due to Richard Parker and many others. They have the disadvantage that they are not guaranteed to succeed on every hyperbolic group presentation, but when they do they are generally much faster than KBMAG. They can also sometimes be carried out by hand.

17.30 Wine reception & book launch (Staff Club in old Quadrangle)

Friday 19 May

10.00‒10.45 Łukasz Grabowski Characterizing amenable groups via measurable variants of the Lovász Local Lemma Lovász Local Lemma is one of the most fundamental tools of combinatorics. It allows finding a colouring of a finite graph with desirable properties. I will describe joint work with E. Csoka, A. Mathe, O. Pikhurko and K. Tyros in which we develop measurable and Borel versions of the local lemma for graphings, i.e. infinite graphs which naturally arise from Borel actions of countable groups. As it turns out, whether or not the Local Lemma holds on a given graphing is very closely related to the amenability of the group which acts.

10.45‒11.30 Coffee/tea

11.30‒12.15 Murray Elder Solving equations in virtually free groups This is joint work with Volker Diekert, Stuttgart. Solving equations in virtually free groups reduces to the problem of solving twisted equations in free groups. Building on previous work with Laura Ciobanu we give a PSPACE algorithm to find all solutions and express them as a formal language of surprisingly low complexity - EDT0L. I will try to give a sketch of our method and show how twisting makes everything considerably harder (or more fun).

12.30‒13.15 Karel Dekimpe The $R_\infty$-property for nilpotent and solvable quotients of groups Any endomorphism $\varphi$ of a group $G$ determines an equivalence relation on $G$ by setting $x \sim y \Leftrightarrow \exists z \in G:\; y= zy\varphi(z)^{-1}$. The equivalence classes of this relation are called Reidemeister classes or twisted conjugacy classes and their number is denoted by $R(\varphi)$. These Reidemeister classes are also connected to the study of fixed points, where they can be interpreted as fixed point classes. There has been a growing interest in the groups $G$ having the $R_\infty$ property, these are groups for which $R(\varphi)=\infty$ for all automorphisms $\varphi$ of $G$.

Given a group $G$ having the $R_\infty$ property, we define the $R_\infty$-nilpotency degree of G as the least least integer $c$ such that the group $G/\gamma_{c+1}(G)$ has the $R_\infty$ property, if such a $c$ exists. Here $\gamma_i(G)$ stands for the $i$-th term of the lower central series of $G$. Analogously, the $R_\infty$-solvability degree is defined. In this talk we discuss these degrees in case $G$ is a free group, a surface group or a Baumslag‒Solitar group.

13.15‒14.30 Lunch

14.30‒15.15 Juan Souto Surface groups acting on the interval I will discuss how to construct smooth actions with free orbits of surfaces groups on the closed interval and other related topics.

15.30‒16.15 Damian Osajda Group cubization I will present a procedure of "group cubization". It transforms a given finitely generated group into a group acting without fixed points on a $\mathrm{CAT}(0)$ cubical complex. As a main application I will provide a family of Burnside groups (that is torsion groups with bounded exponent) without Kazhdan's property (T).

16.15‒17.00 Coffee/tea

17.00‒17.45 Martin Newell Random thoughts

19.00 Conference Dinner (The House Hotel)

Saturday 20 May

10.00‒10.45 Péter Pálfy Elliptic curves and finite $p$-groups The famous PORC conjecture of Graham Higman states that for any fixed $n$ the number of groups of order $p^n$ should be given by polynomials in $p$ for each residue class modulo some $N(n)$. It has been verified up to $n=7$. Recent results of Marcus du Sautoy and Michael Vaughan-Lee, and also by Seungjai Lee point towards the possiblity that the PORC conjecture may not be true perhaps for $n=10$ or even $n=9$. At the heart of these results lies the group of order $p^9$ constructed by du Sautoy in 2002, that encodes in some way the elliptic curve $y^2=x^3-x$.

Another conjecture of Higman from 1960 claims that for any fixed $n$ the number of conjugacy classes in the group of the $n$-by-$n$ upper unitriangular matrices over the $p$-element field is a polynomial in $p$. With Zoltán Halasi we constructed certain subgroups (so-called pattern groups) of the group of unitriangular matrices such that their class number is not a polynomial function in $p$. In this construction we also used the same elliptic curve. Although du Sautoy's group cannot be represented as a pattern group, it can be given as an equa-pattern group contained in the group of $13$-by-$13$ unitriangular matrices. Then its class number can be calculated easily, and it is also not a polynomial in $p$, as it has already been observed by du Sautoy and Vaughan-Lee.

10.45‒11.15 Coffee/tea

11.15‒12.00 Eric Swenson Two shall be the number of the counting In ancient times, Brian Bowditch showed that if the boundary of a one-ended hyperbolic group has a local cut point $p$, then $p$ is part of a cut pair, and the group virtually splits over virtually $\mathbb{Z}$.

We show that if the boundary of a one ended $\mathrm{CAT}(0)$ group $G$ is separated by a finite subset then it is separated by a cut pair, and $G$ virtually splits over virtually $\mathbb{Z}$. We also examine nesting actions on $\mathbb{R}$-trees (because they wouldn't get out of the way).

12.15‒13.00 Arnold Feldman System permutability and nilpotent length This report describes continuing collaboration with Rex Dark and Mariá Dolores Pérez-Ramos.

A significant measure of the tractability of a Fitting class is whether its injectors are system permutable. In recent work, we began to detail properties of finite soluble groups that are in some sense minimal with respect to having injectors of a given Fitting class that are not system permutable; these properties were used to prove that the injectors of certain Fitting classes are always system permutable.

In this work we deal with arbitrary Fitting classes but a specific property of soluble groups: nilpotent length. After sharpening some results regarding the minimal counterexamples mentioned above, we show first that in every group of nilpotent length less than $5$ the injectors for every Fitting class are system permutable. We then describe a previously unpublished example constructed by Dark consisting of a group $H$ with nilpotent length $5$ along with a Fitting class $\mathcal X$. We then outline how Lockett modified $\mathcal X$ to obtain a Fitting class $\mathcal F$ whose injectors in $H$ are not system permutable, thereby verifying that our nilpotent length result is the best possible.

Registration

There is no registration fee but for making dinner arrangements and pleasing our generous sponsors it would be nice if anybody who wants to participate and join the conference dinner could please send an email to Dieter Degrijse (dieter dot degrijse at nuigalway dot ie) with subject line "gig2017 registration", indicating your name, affiliation and whether you want to join the conference dinner, including special dietary requirements if applicable.

Support

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 your CV by 20 April 2017.

Travel

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 up 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.

Accommodation

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 Dieter Degrijse and Claas Röver.

Groups in Galway 2017 is generously supported by