Winter One-Relator Workshop, 9th-11th January 2018
School of Mathematics, University of East Anglia, Norwich
This workshop is took place as part of the EPSRC project EP/N033353/1 "Special inverse monoids: subgroups, structure, geometry, rewriting systems and the word problem". The workshop brought researchers with expertise on one-relator groups together with researchers with expertise in the theory of one-relator monoids, and inverse monoids, to present their results and discuss possible future research directions.
The workshop will begin on Tuesday 9th January at 14h00 and end on Thursday 11th January at 13h00.
Arrival and lunch on Tuesday: For those who will arrive in time for lunch on Tuesday, I propose that we meet at 12h30 in the UEA central Square (outside the Campus Kitchen building). Then we can go together to the university canteen
for lunch. Detailed instructions on getting to UEA and the Square may be found in the
Visitor's guide to Norwich and the UEA.
Location for the talks
The talks will be held in the School of Mathematics of the University of East Anglia in room SCI 3.05. To find this room, you enter the maths building and then make your way up to the top floor.
For more information about how to get to the maths department at UEA please see the following guide (thanks for Tom Coleman for putting this together):
Visitor's guide to Norwich and the UEA
- Laura Ciobanu (Heriot-Watt, UK)
- Igor Dolinka (Novi Sad, Serbia)
- Giles Gardam (Technion, Israel)
- Jim Howie (Heriot-Watt, UK)
- Alan Logan (Glasgow, UK)
- Lars Louder (UCL, UK)
- Stuart Margolis (Bar Ilan, Israel)
- John Meakin (Nebraska-Lincoln, USA)
- Nik Ruskuc (St Andrews, UK)
- Nora Szakács (Szeged, HUN)
If you are interested in attending the workshop then please send me an email. There is no registration fee.
We will go for dinner on the evening of Wednesday 10th January. If you would like to join for dinner and have not already told me, then please let me know by email.
Timetable of talks
Talk titles and abstracts
- Extended Dehn algorithms and the generalised word problem
Laura Ciobanu (Heriot-Watt, UK)
Abstract: In this talk I will introduce an extension of Dehn's algorithm due to Cannon, Shapiro and Goodman, and explain how with additional conditions it can lead to solving the word problem in real time. I will then apply these ideas to solve the generalised word problem, that is, deciding whether a word on the group generators belongs to a particular subgroup, in (relatively) hyperbolic groups. This is joint work with Derek Holt and Sarah Rees.
Solving the word problem of the O'Hare monoid far, far away from sweet home Chicago
Link to abstract.
Igor Dolinka (Novi Sad, Serbia)
- Finding and not finding unusual one-relator groups
Giles Gardam (Technion, Israel)
Abstract: This talk has two parts. First, I'll present joint work with Daniel Woodhouse in which we construct one-relator groups that are virtually Brady-Bridson groups. They cannot act on CAT(0) cube complexes, and their Dehn functions are polynomials with exponents dense in [2, \infty), so in particular they give a negative answer to the question of whether Baumslag-Solitar subgroups are the only obstruction to automaticity for one-relator groups. Second, I'll outline my ongoing computational project to build a census of small one-relator groups, and the unsuccessful search for a counterexample to the automaticity question.
- One-relator groups and one-relator products
Jim Howie (Heriot-Watt, UK)
Abstract: A one-relator product of groups is the quotient of a free
product by the normal closure of a single element. In particular, a
one-relator group is a one-relator product of infinite cyclic groups.
So one-relator products are natural generalisations of one-relator
groups. I will survey some older results on one-relator groups and on
one-relator products, then report on some more recent joint work with
- One relator groups with torsion
Alan Logan (Glasgow, UK)
Abstract: One relator groups with torsion are hyperbolic. In this talk we will explore how (and, I claim, why) one relator groups with torsion bridge the gulf in difficulties between dealing just with torsion free hyperbolic groups and dealing with all hyperbolic groups.
- Negative immersions for one-relator groups
Lars Louder (UCL, UK)
Abstract: Lyndon's theorem that a commutator in a free group is not a proper
power, Magnus' Freiheitssatz, Wise's W-cycles conjecture, Stallings'
theorem that injections of free groups inducing injections on
abelianization are injective on the set of conjugacy classes, and Baumslag's
theorem on adjoining roots to subgroups of free groups, are all special
cases of a single dependence theorem. As a corollary there is a new
family of one-relator groups without Baumslag-Solitar subgroups, and a
criterion for k-freeness of a one-relator group.
This is joint work with Henry Wilton.
- Inverse Semigroups for the Working Algebraist
Link to abstract.
Stuart Margolis (Bar Ilan, Israel)
- The word problem for one relator inverse monoids: techniques and obstacles
Link to abstract.
John Meakin (Nebraska-Lincoln, USA)
- On groups of units in one relation monoids and inverse monoids
Nik Ruskuc (St Andrews, UK)
Abstract: In his seminal 1966 paper Adjan proved that the word problem for one relation special monoids, defined by a single relation of the form u=1, is decidable. The key in proving this is to establish a presentation for the group of units for such a monoid, which turns out to be a one-relator group, with a relator obtained by 'reading' u in certain 'pieces'. Zhang 1992 provided a very insightful reworking of this result in the language of rewriting systems. In the world of one relation special inverse monoids, an analogous generating set, consisting of `pieces' of u was established by Ivanov, Margolis and Meakin (2001). They also show that if the word problem is decidable for one relator special inverse monoids, then it is decidable for all one relator monoids (and semigroups). However, going from the generating set established by IMM to a presentation for the group of units has remained elusive since then. It is not even known whether the group of units is necessarily finitely presented, let alone one-relator. In my talk, I will attempt to outline the landscape presented by these three papers in more detail and present some relevant recent insights obtained by R. Gray and myself. I will then try to initiate and facilitate a discussion among the participants about these observations by posing some open problems that seem to suggest themselves as a consequence. These will range from problems entirely to do with one-relator groups, via computability questions regarding the generating set, and the speculations of how one would go from a presentation and the word problem for the group of units (elusive as it is at present) to an attempt to treat the word problem.
- Two examples of geometric methods in inverse semigroup theory
Link to abstract.
Nora Szakács (Szeged, HUN)