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 cut points in the boundary of relatively hyperbolic groups. His comments in the page 64 of the following monologue is particularly interesting.

Bowditch says that if the boundary of a relatively hyperbolic group is connected then each of its global cut points is a parabolic fixed point.

“For this one need to place certain mild restrictions on the class of groups that can occur as maximal parabolic groups. (It is sufficient to assume that they are one or two-ended, finitely presented, and not infinite torsion groups. Probably only the last of these assumptions is really important.)”

Do we really need all three of these conditions? That is, do the maximal parabolic subgroups need to be

  • one or two ended
  • finitely presented
  • not infinite torsion group (Bowditch thinks only this one is really important)

In the same monologue he refers to Swenson’s paper which has an ‘alternative route’.

Bestvina’s paper hints at a route that may get rid of these extra conditions.

There are three components of his thought:

  1. relatively hyperbolic group
  2. boundary of a group
  3. cut points

We will try to explore each of these ideas.

Published by Ashani Dasgupta

Pursuing Ph.D. in Geometric Group Theory at University of Wisconsin, Milwaukee

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