
Geogebra model for hyperbolic isometries
The goals of this GeoGebra model are the following: 1) Input a hyperbolic isometry of the hyperbolic plane using the matrix representation. In other words input a real matrix with determinant 1 and trace more than 2. 2) Programmatically draw the axis. 3) Mark any point P on the hyperbolic plane (upper half plane) 4) […]

Protected: Poetry of Category Theory
There is no excerpt because this is a protected post.

Higher Dimensional Kleinian Groups
My advisor says, I should invest a portion of my time, to something disjoint from my research area (= local connectedness of the boundary of relatively hyperbolic groups). Higher dimensional Kleinian groups are not exactly disjoint from that. Nevertheless they are somewhat of a different flavor. Might I say, more concrete, that the spiraling labyrinth […]

The invariant measure (reading / replicating parts of the paper by Levitt)
From K to L \( \mathcal{K} = \{\phi_i :A_i \to B_i \}_{i=1, … , k} \) be a nonnesting closed system on a finite tree with at least one infinite orbit. Non nesting forces a special structure for the set of finite regular \( \mathcal{K} \) – orbits Union of all finite regular orbits = […]

Protected: Tameness 1
There is no excerpt because this is a protected post.

Dunwoody – Day 6
A JSJ decomposition (or JSJ tree) of G over A is an Atree T such that: T is universally elliptic; T dominates any other universally elliptic tree T’. An Atree is universally elliptic if its edge stabilizers are elliptic in every Atree. Recall that H is elliptic in T if it fixes a point in […]

Dunwoody’s accessibility theorem – Talk Day 4
This is a personal musing. Possible errors, uncredited excerpts lie ahead. We constructed sequence of equivariant maps \(f_k\) from the universal cover \( \tilde {X} \) to the sequence of refinements \(T_k\). The construction was complete up to the 1skeleton. We want to extend the maps to 2skeleton in a certain way. To motivate the […]

Cut points in Bowditch Boundary of Relatively hyperbolic groups 2
Understanding Swenson (large excerpt.. some diagrams .. some remarks). Please be cautious. Potentially wrong remarks lie ahead. Continuum A continuum is a compact connected Hausdorff space. Cut Point In a continuum Z, \( c \in Z \) is a cut point if \( Z = A \cup B \) where A and B are nonsingleton continua and \( A \cap B = \{c\} \). If in addition \( D \subset […]

Dunwoody’s Accessibility Theorem – Day 2
Suppose G is a finitely presented group. Let us fix \( \mathcal{A} \) – a favorite class of subgroups of G (closed under taking subgroups and conjugation). If G acts on an \(\mathcal{A} \) – tree T, we have a graph of groups decomposition for G. If H < G then H acts on T […]

Cut points in Bowditch Boundary of Relatively hyperbolic groups 1
This document is a personal musing. It has many excerpts without credit, potentially false claims, and misquotes. If some cosmic accident has lead you to this page, then take a deep breath and assume caution. If you are worried about copyright infringement, kindly let me know. I will modify the document. B.H. Bowditch thought about […]