# Mathematics. Teaching. And Epsilon.

• ## 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 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. […]

• ## 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 […]

• ## 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.

• ## Round robin tournament

Problem : Suppose there are teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the team loses games and wins games.Show that = Solution : Each team plays exactly one match against each other team. Consider the expression Since each team […]

• ## Orthogonality

Let ABC be a triangle and D be the midpoint of BC. Suppose the angle bisector of $\angle ADC$ is tangent to the circumcircle of triangle ABD at D. Prove that $\angle A = 90^o$ . (Regional Mathematics Olympiad, India, 2016)

• ## Construction of polynomials

The polynomial P(x) has the property that P(1), P(2), P(3), P(4), and P(5) are equal to 1, 2, 3, 4, 5 in some order. How many possibilities are there for the polynomial P, given that the degree of P is strictly less than 4? (Duke Math Meet 2013 Tiebreaker round) Discussion: Let \[P(x) = a […]

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