Non-positive Curvature in Groups
Abstract: In this document we explore some of the relationships between different classes of groups important to geometric group theory and different properties they enjoy. These classes often coincide and the properties can have non-trivial intersection. Looking at these intersections, i.e. at groups having multiple properties of interest, we notice even stronger properties that come as a result. ... read moreIn particular we characterize the intersection of acylindrical hyperbolicity with CAT(0) and apply this classification to right angled Coxeter groups. Some other classes we are particularly concerned with are those of the braid groups and automorphism groups. Specifically, in the second class, we will look at automorphisms of free groups, as well as automorphisms of the Universal Right Angled Coxeter groups, better known as the free product of a finite number of copies of groups of order 2. Both these types of groups have had questions raised about them that we will address, in particular as to whether they are CAT(0). In exploring the relationship between these classes, and the relatively new notion of Acylindrical Hyperbolicity, we get some interesting results pertaining to these questions. This comes from discovering that the braid groups (modulo a cyclic subgroup) and the automorphisms of our universal groups (modulo inner elements) satisfy this notion of acylindrical hyperbolicity, which can be seen as a very generalized version of negative curvature. In doing this, we will discover that, should either of these classes of groups prove to be CAT(0), then they must act on a specific kind of CAT(0) space, particularly a space which is `rank one'. Because we know that sufficiently low index braid groups are indeed CAT(0), we explore such a space it acts on geometrically, and distinguish exactly such a `rank one' element which we now know must exist. We continue on to explore more about these acylindrical actions generally and their relationship to relative hyperbolicity. Both hyperbolicity and relative hyperbolicity exhibit a nice characterization of limit sets of appropriate actions in the form of boundary. We demonstrate why an analogous structure doesn't appear for acylindrically hyperbolic groups. In doing so, we explicitly construct two actions, both universal in the appropriate sense, that have markedly different end behavior. This leads us towards a desire to classify the rigidity present in these classes, and how it weakens as we relax the hypotheses on the action of interest. In doing so, we prove a folklore result about relatively hyperbolic groups and neatly summarize the rigidity of these group actions.
Thesis (Ph.D.)--Tufts University, 2018.
Submitted to the Dept. of Mathematics.
Advisor: Genevieve Walsh.
Committee: Kim Ruane, Ruth Charney, and Robert Kropholler.
Keyword: Mathematics.read less