The Geometry of Amalgamations

Abstract:  As a geometric group theorist, we study infinite groups by viewing them as symmetry groups of a space - I.e. isometries of a metric space, automorphisms of a combinatorial object like a graph, or homeomorphisms of a topological space. In this way, we try to “visualize” algebraic properties of the group.  In this talk, I want to discuss how a geometric group theorist “see” the structure of amalgamated products and HNN extensions through this lens.  As an example of what I mean, there is a well-known theorem of Papasoglu’s that says, a finitely presented group G (that is not virtually Fuchsian) splits as an amalgam over a 2-ended subgroup if and only if there is a quasi-line that separate any Cayley graph of G.