SBCW-Complexes and n-Cost

Explore the mathematical foundations of measure-preserving standard Borel CW complexes and their applications to group theory in this 54-minute conference talk. Delve into the concept of discrete standard Borel groupoids as internal groupoids in the category of standard Borel spaces and Borel maps with countable fibers. Learn how these structures, when equipped with preserved Borel measures, form measure-preserving standard Borel groupoids (m.p. SBG). Discover the introduction of measure-preserving standard Borel CW complexes (m.p. SBCW complexes) as an analogue to the relationship between algebraic topology and group theory. Examine how every m.p. SBG admits a classifying space that is unique up to homotopy equivalence. Understand the definition and calculation of n-dimensional truncated Euler characteristics of m.p. standard Borel groupoids through the infimum of Euler characteristics over all classifying spaces. Investigate the n-dimensional cost of countable groups defined as the infimum of n-dimensional truncated Euler characteristics of associated action groupoids from probability measure-preserving actions. Analyze the connections between n-dimensional cost, n-dimensional rank gradient, and ℓ²-Betti numbers of countable groups through collaborative research with Miklós Abért and Damien Gaboriau.