# Category: Uncategorized

• ## 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 non-nesting 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 = […]

• ## Dunwoody – Day 6

A JSJ decomposition (or JSJ tree) of G over A is an A-tree T such that: T is universally elliptic; T dominates any other universally elliptic tree T’. An A-tree is universally elliptic if its edge stabilizers are elliptic in every A-tree. 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 1-skeleton. We want to extend the maps to 2-skeleton in a certain way. To motivate the […]

• ## Day 3 (notes from Craig’s Lecture)

Definition A subset N of a space X is a neighborhood of infinity if $$\bar{X /N }$$ is compact. We say that X has k ends ( $$k \in \mathbb{N} \cup \{ \infty \}$$ ) if \( k = sup \{ j | X \textrm{has a nbd of} \infty \textrm{with j […]

• ## Free Groups, Gromov Hyperbolicity – (translated excerpt from Harpe)

Soit T un arbre simplicial muni d’une distance pour laquelle chaque arête est isométrique au segment [0,1] de la droite réelle, et pour laquelle la distance entre deus points est la borne inférieure des longueurs des chemins joignant ces points. Tout triangle de T est dégénéré au sens oú chacun de ses côtés est contenu […]