Category: Geometric Group Theory

Game of Mario in Heisenberg group
Yesterday we were discussing the Heisenberg group in our weekly Geometric group theory workshop. The game of Mario came up! If you have not played Mario, then here is a the only thing you need to know. Mario can jump and hit floating bricks. These bricks may pop open to produce gold coins. The method…

Not far not near groups
We use \(NFNN\) as an abbreviation for ‘not far not near’. Annulus \(A(w, r, R)\) in a metric space \((X, d)\), for \(w \in X, 0 < r < R < \infty \) is defined as the set \[A =\{ x \in X  r \leq d(w, x) \leq R \} \]. Definition Suppose \(P\)…

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 1skeleton. We want to extend the maps to 2skeleton in a certain way. To motivate the…

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 nonsingleton continua and \( A \cap B = \{c\} \). If in addition \( D \subset…

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…

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 inclass 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 2complex). 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.…

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