Investigating local connectedness of boundary of relatively hyperbolic groups - 1

This is an update from Cheenta Research Track (Geometric Group Theory group).Here, we will investigate local connectedness of boundary of hyperbolic groups. The group is comprised of Ashani Dasgupta, Sambuddha Majumdar. Learn more about Research Track here.

Reference Texts:

• Metric Spaces of Non-Positive Curvature by Haefliger
• Algebraic Topology by Hatcher
• Contemporary Abstract Algebra by Gallian
• Aspects of Topology by Charles O. Christenson et. al

Reference Papers

• R - trees in topology, geometry, and group theory by Mladen Bestvina

Apology

This is a close reading of developments in Geometric Group Theory. I have learnt most of it from my doctoral advisor. In this work group, we are investigating some aspects of this vast field.

Sketch of discussion (till 13th May, 2019)

Suppose X is a set. Also assume that there is a rule to measure distance between points in the set. This is loosely what a metric space is. (Read more about metric spaces and point set topology in Christenson)

Isometries are maps from X to X which preserves distance. Suppose A and B are two points in X. Then, if distance between A and B is same as f(A) and f(B), then f is known as an isometry

|A-B| = |f(A) - f(B)|

We consider the set of all isometries of X. This is a group (learn more about groups from Gallian).

Proper Action

Let \( \Gamma\) be a group acting by isometries on a metric space X. The action is said to be proper (alternatively, "\( \Gamma \) acts properly on X " ) if all but finitely many members of \( \Gamma \) move small enough balls about each point disjointly from the ball.

We went ahead and proved the Proposition 8.5 from Haeflegar rigorously.

Suppose a group \( \Gamma \) acts properly by isometries on the metric space X. Then:

(1) For each \( x \in X \), there exists \( \epsilon > 0 \) such that if \( \gamma \cdot B (x, \epsilon ) \cap B (x, \epsilon) \neq \phi \) then \( \gamma \in \Gamma_x \), the stabilizer of x.

(2) The distance between the orbits in X defines a metric on the space \( \Gamma \backslash X \) of \( \Gamma \) orbits.

(3) If the action is proper and free, then the natural projection \( p : X \to \Gamma \backslash X \) is a covering map and a local isometry.

(4) If a subspace Y of X is invariant under the action of a subgroup \( H \subseteq \Gamma \) then the action of H on Y is proper.

(5) If the action of \( \Gamma \) is cocompact then there are only finitely many conjugacy classes of isotropy subgroups in \( \Gamma \)

The discussion involved rigorous proofs and definitions of relevant terms. We backtracked inside some ideas from topology (covering space theory) and group theory (isotropy groups, conjugacy classes).

What lies ahead

So, we discussed hyperbolic groups. We will review the covering space theory in some detail and finally understand the Schwarz - Milnor theorem (notion of quasi - isometry).