Workshop Schedule K60: Groups and Cohomology

Workshop Schedule, K60: Groups and Cohomology

All lectures will take place in the Lecture room 4A of the Mathematics Building, which is building 54 on the top left of this map.

Tea, coffee and a reception on Monday evening will be held in Building 56 on the same map.

For lunch, participants are recommended to visit the Staff Social Club, in Building 38 near the centre of the map.

Monday 19th March
09:00-10:00 Registration
10:00-11:00 Tara Brendle
Coffee
11:30-12:30 Ioannis Emmanouil
Lunch
14:30-15:30 Brita Nucinkis
Tea
16:30-17:30 Nikolay Nikolov
18:00-19:00 Wine Reception

Tuesday 20th March
09:30-10:30 Brendan Owens
Coffee
11:15-12:15 Conchita Martínez-Pérez
Lunch
14:00-15:00 Olympia Talelli
Tea
16:00-17:00 Martin Bridson
19:00 Conference Dinner at Ceno Restaurant.

Wednesday 21st March
09:30-10:30 Aditi Kar
Coffee
11:15-12:15 Alan Reid

Titles and Abstracts

1. Tara Brendle (Glasgow): Normal subgroups of mapping class groups

Abstract: The mapping class group of a surface has an incredibly rich normal subgroup structure. For this reason, a traditional classification theorem for normal subgroups of mapping class groups, in the form of a complete list of isomorphism types, is almost certainly out of reach. However, we might hope to gain some insight via certain invariants of normal subgroups. In this talk, we will view the automorphism group as an invariant of normal subgroups, and give a survey of known examples, providing evidence for a conjectured classification of normal subgroups based on recent joint work with Dan Margalit.

2. Ioannis Emmanouil (Athens): Flatness versus projectivity for modules and complexes

Abstract: We will present a new criterion for the finiteness of the projective dimension of flat modules over group rings. We will also analyze the relation between K-flat and K-projective complexes, as defined by Spaltenstein.

3. Brita Nucinkis (Royal Holloway): Classifying spaces for families and their finiteness conditions

Abstract: I will give a survey on cohomological finiteness conditions for classifying spaces for families of subgroups, such as the dimension or the type and will discuss some old and new questions.

4. Nikolay Nikolov (Oxford): On conjugacy classes in compact groups

Abstract: It is well known that if G is a finite group with k(G) conjugacy classes, then k(G) tends to infinity as |G| tends to infinity. With Andrei Jaikin-Zapirain we prove that if G is an infinite compact Hausdorff group then k(G) is uncountable. The proof easily reduces to the case of profinite groups and uses results on finite groups with an almost regular automorphism.

5. Brendan Owens (Glasgow): Slice surfaces and double-branched covers

Abstract: Gordon and Litherland defined a bilinear form on the first homology of an embedded surface in the 3–sphere, and used it to give a formula for the signature of the bounding link. I will describe a generalisation of this to embedded surfaces in the 4–ball. I will also discuss how this may be useful in searching for ribbon alternating knots. This is based on joint work with Greene and Strle, and also joint work with Swenton.

6. Conchita Martínez-Pérez (Zaragoza): On automorphisms groups of RAAGs

Abstract: We consider several problems on automorphisms groups of RAAGs. First, we consider the case when the defining graph is a tree and study from a combinatorial perspective the value of the first Betti number of certain finite showing that as the number of vertices grows, the proportion of trees for which that Betti number is zero tends to zero. Then, we look at a different subgroup of the full automorphism group: the subgroup generated by partial conjugations. We construct a normal subgroup of this group which is a RAAG itself and use this construction to give a partial characterization of the cases when the associated Lie algebra is Koszul. This is a joint work with Javier Aramayona, José Fernández, Pablo Fernández aand Luis Mendoça.

7. Olympia Talelli (Athens): On the Gorenstein Dimension of Groups

Abstract: The Gorenstein cohomological dimension of a group generalizes the ordinary cohomological dimension of the group in the sense that the two coincide when the latter is finite and it is related to many other invariants that are studied in cohomological group theory. We'll see that it shares many properties with the cohomological dimension and it appears in the study of groups with periodic cohomology as well as in the study of groups which admit a finite dimensional classifying space for proper actions. We'll also see its relation to the cohomology of the group with group ring coefficients.

8. Martin Bridson (Oxford): Direct products, weak commutativity and finiteness properties of groups

Abstract: I shall begin by discussing how to construct concise presentations for direct products of groups, and how difficult it is to determine the finiteness properties of subdirect products. I shall then present joint work with Dessislava Kochloukova concerning the finiteness properties of Sidki doubles X(G), i.e. groups obtained from a free product G*G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. Deceptively complicated finitely presented groups arise from this construction: X(G) is finitely presented if and only if G is finitely presented, but if F is a non-abelian free group, the X(F) has a subgroup of finite index whose third homology is not finitely generated.

9. Aditi Kar (Royal Holloway): 2D Problems in Groups

Abstract: I will discuss a conjecture about stabilisation of deficiency in finite index subgroups and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem for group presentations. We can prove the pro-p version of the conjecture, as well as its higher dimensional abstract analogues. Key ingredients are, first a classic result of Wall on the existence of CW complexes with prescribed cellular chain complex, and second, a simple criterion for free-ness of modules over group rings. This is joint work with Nikolay Nikolov.

10. Alan Reid (Rice): The topology of arithmetic hyperbolic 4-manifolds

Abstract: Suppose that M is a smooth closed oriented 4-manifold. The intersection form Q_M on H_2(M,Z)/Tor(H_2(M,Z)) is an important topological invariant of M. This talk will describe some recent work on some topological questions regarding hyperbolic 4-manifolds; for example the question of which possible intersection forms occur for closed arithmetic hyperbolic 4-manifolds.

Last updated March 9th 2018.