Jacek Świątkowski (University of Wrocław)

Title: Topological characterization of boundaries of free products of groups.

Abstract: I will introduce the operation, called dense amalgam, which to any tuple X1,..., Xk of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted ⨆ {\displaystyle \bigsqcup ⊔(X1, . . . , Xk), in which there are many appropriately distributed copies of the spaces X1, . . . , Xk. I will also present a convenient characterization of dense amalgams, in terms of a list of properties, similar in spirit to the well known characterization of the Cantor set. I will explain that, in various settings, the ideal boundary of the free product of groups (amalgamated along finite subgroups) is homeomorphic to the dense amalgam of boundaries of the factors. For example, the boundary of a Coxeter group which has infinitely many ends, and which is not virtually free, is the dense amalgam of the boundaries of the maximal 1-ended special subgroups.