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# Category Archives: Geometric Group Theory

## Rips Machine

Let G be a finitely presented group acting minimally, stably and non trivially by isometries on an \( \mathbb{R} \) tree S. If G does not split over an arc stabilizer of S, then one of the following is true: There is a line \( L \subset S \) acted on by a subgroup H […]

## 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 […]

## 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 […]

## 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. […]

## 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 […]

## 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 […]

## 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).

## 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 […]