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… Continue reading Higher Dimensional Kleinian Groups

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 =… Continue reading The invariant measure (reading / replicating parts of the paper by Levitt)

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… Continue reading Dunwoody – Day 6

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… Continue reading Dunwoody’s accessibility theorem – Talk Day 4

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 non-singleton continua and \( A \cap B = \{c\} \). If in addition \( D \subset… Continue reading Cut points in Bowditch Boundary of Relatively hyperbolic groups 2

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… Continue reading Free Groups, Gromov Hyperbolicity – (translated excerpt from Harpe)