North British Geometric Group Theory Seminar

Speakers

• Bianca Marchionna (UC Louvain)

• Ric Wade (Warwick)

• Richard Webb (Manchester)

Titles and Abstracts

Bianca Marchionna:  Topological right-angled Artin groups

Right-angled Artin groups (RAAGs) form a family of finitely generated groups that play an important role in geometric and computational group theory. Their study is often related to the one of their universal Salvetti complex, a CAT(0)-cube complex on which they act geometrically.

In this talk, we introduce a generalisation of RAAGs to topological groups. Remarkably, those groups have a cellular action on a thicker version of the universal Salvetti complex of a RAAG with controlled cell stabilisers. Due to the high connectivity of this complex, we can deduce homological/homotopical finiteness properties for topological RAAGs. 

Work in progress with I. Castellano, B. Nucinkis, and Y. Santos Rego.

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Ric Wade: Distortion and coarse median preserving automorphisms of right-angled Artin groups 

Alibegović showed that every outer automorphism of a free group has positive translation length in any Cayley graph of Out(F_n). This is equivalent to the corresponding cyclic subgroup being undistorted. She further showed that for a given Cayley graph, this translation length is uniformly bounded below. Combined with work of Conner, this implies that every solvable subgroup of Out(F_n) is virtually abelian.

The family of outer automorphism groups of right-angled Artin groups (RAAGs) includes Out(F_n) but also includes GL(n,Z)  for all n. As a result (for instance by considering the Heisenberg group), we cannot hope for Alibegović’s theorem to apply to every Out(RAAG). Charney, Stambaugh, and Vogtmann introduced a subgroup of what they call untwisted automorphisms, which roughly speaking behaves more like Out(F_n) than GL(n,Z). This was later shown by Fioravanti to coincide with the group of automorphisms preserving the coarse median structure on the RAAG induced by the Salvetti complex. We use this characterisation to generalise Alibegović’s theorem to the untwisted automorphism group of a RAAG. The description of these automorphisms as coarse median preserving is key - it allows one to extend arguments for free groups that use the so-called `bounded backtracking property’ to arbitrary special groups. Uniformity is still slightly out of reach, however we can prove this with the addition of an extra hypothesis (which we think is reasonable enough to hold in general). Based on work in progress with Jerónimo Garcia-Mejía.

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Richard Webb: Equators of the two-sphere and area-preserving homeomorphisms 

We'll discuss a new way of studying the group of area-preserving homeomorphisms of the two-sphere, using geometric group theory. Although the sphere is simply connected, it turns out that one can use the area (instead of appealing to fundamental groups) to construct analogues of the curve graph (which have been important in the study of mapping class groups). As an application, we show that certain metrics on the space of "equators" of the sphere have infinite diameter, and discuss how this work relates to a problem from symplectic geometry, namely the Equator Conjecture. Joint work with Yongsheng Jia. 

Social dinner

We will have a social dinner at HOME. If you plan to attend, please complete the following form by latest 1st March. Funding will be available to cvoer part of the dinner costs for ECRs. 

Travel support

There is some travel funding available for PhD students and ECRs. If you are interested in travel funding, please contact Gemma Crowe and Alexandre Martin before the meeting.

Seminar information

The North British Geometric Group Theory Seminar is a collaborative seminar that has been running since 2003. The seminar involves geometric group theorists from Aberdeen, Heriot–Watt, Glasgow, Newcastle, Durham, York, Leeds, Manchester, Nottingham, St Andrews, and Leicester, and meets three times a year.

We are very grateful for financial support from the London Mathematical Society, the Glasgow Mathematical Journal Learning and Research Support Fund and the Heilbronn Institute for Mathematical Research.