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A survey of relative Dunwoody’s accessibility theorem

Motivation This is not (even remotely) an original work. For example it contains large excerpts from a variety of papers (often without reference). More importantly beware! What follows may contain outrageously false statements. This was created for an in-class presentation while the author was exploring these ideas for the first time.. Consider a group G […]

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Accessibility

G is a finitely presented group. X is its presentation complex (a simplicial 2-complex). Since G is finitely presented, the number of vertices of X is finite. Suppose $$u_1 , \cdots , u_q$$ be the vertices of X. $$\tilde {X}$$ be its universal cover. Fix lifts of the vertices of X. […]

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Cut points in Boundary – 1

Proceedings from Top Secret Topology Seminar. Disclaimer: Handle with care. Several unproved (possibly untrue) statements lie ahead. Jump to summary. Le G be a word-hyperbolic group and $$\partial G$$ its boundary. Theorem (Bestvina, Messa, 1991) If G has one end, then $$\partial G$$ is connected and locally connected. The second part […]

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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 […]

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একটি সূত্রের জন্ম

(ফরাসি গণিতজ্ঞ, সেড্রিক ভিলানি, ‘থিয়োরেম ভিভান্ত’ গ্রন্থ রচনা করেন ২০০৮ সালে। বইটি খানিকটা ডায়রি লেখার ঢঙে রচিত। মূল ফরাসি থেকে বাংলায় অংশ বিশেষ অনুবাদ করছি স্রেফ খেলার ছলে। কোনো বাণিজ্যিক উদ্দেশ্য নেই।)।  লিয়ঁ, ২৩শে মার্চ, ২০০৮ রবিবার দুপুর ১টা। প্রায় জনহীন গবেষণাগারে দু’জন ব্যস্ত গণিতজ্ঞ ছাড়া বোধকরি আর কেউ নেই। লিয়ঁর এই গবেষণাকেন্দ্রে আমি গত […]

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The Alexander Trick

Here is the original paper: J. W. Alexander, On the deformation of an n-cell (A 2-page paper that influenced a remarkable amount of later work).

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Ends

Motivation Start with a locally finite simplicial complex X. Locally finite: Each vertex is attached to only finitely many simplices. Why locally finite: To make sure it is a CW complex. Notice that the closure-finite criteria require each cell of a CW complex to meet only finitely many other cells. Hence we do not have a situation like […]

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Which manifold is this?

This is the first exercise from Thurston’s Three Dimensional Geometry and Topology Vol. 1. Which manifold is this? It is like an old trick. Try following the lines. There are actually 6 loops (circles) in this maze. Here is a color coded picture of it.