Author: Ashani Dasgupta
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How to teach mathematics : an experiment with triangular numbers and splitting of plane
Mathematics is all about the beauty of patterns and their reasonable connections. How about connecting patterns from seemingly different domains of the subject? This is a note borne out of a Geometry workshop at Cheenta where we tried exactly that. The audience comprised of 9 to 11 years old students. The purpose of this note…
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Geogebra model for hyperbolic isometries
The goals of this GeoGebra model are the following: 1) Input a hyperbolic isometry of the hyperbolic plane using the matrix representation. In other words input a real matrix with determinant 1 and trace more than 2. 2) Programmatically draw the axis. 3) Mark any point P on the hyperbolic plane (upper half plane) 4)…
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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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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…
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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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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).