Hiroki Ishikura (University of Tokyo)
Title: Stallings-Swan's theorem for Borel graphs, part 1
Abstract: We prove that a Borel graph with uniformly bounded degrees of cohomological dimension one is Lipschitz equivalent to a Borel acyclic graph, and in particular, it is treeable. This is an analog of Stallings-Swan's theorem that a group of cohomological dimension one is free. The proof uses decomposition of Borel graphs which is given by combining Dunwoody's criterion for accessibility of groups and Tserunyan's descripitive construction of structure trees. That is, we show that if a Borel graph with uniformly bounded degrees satisfies a cohomological assumtion, then it is essentially a free product of two Borel graphs, one of which is acyclic and the other is uniformly at most one-ended. In the first talk, we explain the main theorem and how it is deduced from the decomposition theorem. In the second talk, we discuss the construction of structure trees and the proof of the decomposition theorem.
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