This is an update from Cheenta Research Track (Geometric Group Theory group). 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).