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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 […]

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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 = […]

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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 […]

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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 […]

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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 […]

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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 […]

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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 […]

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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 […]

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Mazur Manifold – Class Lecture

Let G = < A | R > be a presentation of a group. Tietze Transformations (finite versions) are ways that a group presentation can be altered without changing the group. \(T_I \) – add a relator that is a consequence of the other relators. example: \(aba^{-1}b^{-1} , aba \) implies \(a^2b \) \(T_{II} \) – add […]

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Genus Two Symmetry

Thurston 1.2.6 (a) Julian Rosen created a photo and answer:  Imagine three lines of longitude on a unit sphere (embedded in ℝ3), at angles 0, 2π/3, and 4π/3. Let G be the union of those three lines of longitude. Then G has three-fold rotational symmetry and the set of points in ℝ3 that are distance exactly […]