Two joint seminars with KIAS HCMC Topology Seminar

• December 11th, 2024 (Wed), 9 am KST

Zoom only (link) | Meeting ID: 819 2513 6908 | Passcode: kias

Mladen Bestvina (University of Utah)

Classification of Stable Surfaces with respect to Automatic Continuity

We provide a complete classification of when the homeomorphism group of a stable surface, Σ, has the automatic continuity property: Any homomorphism from Homeo(Σ) to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of Σ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable second countable Stone space has the automatic continuity property. Under the presence of stability this answers two questions of Mann. This is joint work with George Domat and Kasra Rafi.

• December 12th, 2024 (Thu), 9 am KST

Zoom only (link) | Meeting ID: 819 2513 6908 | Passcode: kias

George Domat (University of Michigan)

Graphical models for topological groups

By analogy with the Cayley graph of a group with respect to a finite generating set or the Cayley-Abels graph of a totally disconnected, locally compact group, we introduce countable connected graphs associated to Polish groups that we term Cayley-Abels-Rosendal graphs. A group admitting a Cayley-Abels-Rosendal graph acts on it continuously, coarsely metrically properly and cocompactly by isometries of the path metric. By an expansion of the Milnor-Schwarz lemma, it follows that the group is generated by a coarsely bounded set and is quasi-isometric to the graph. In other words, groups admitting Cayley-Abels-Rosendal graphs are topological analogues of the finitely generated groups. We will see these concepts in action by considering homeomorphism groups of countable Stone spaces (i.e. homeomorphism groups of countable end spaces/countable ordinals). We completely characterize when these homeomorphism groups are coarsely bounded, when they are locally bounded (all of them are), and when they admit a Cayley-Abels-Rosendal graph, and if so produce a coarsely bounded generating set. This is joint work with Beth Branman, Hannah Hoganson, and Robbie Lyman.