[Binghamton Geometry Topology Seminar] This week: Elizabeth Field

[jwilliams at math.binghamton.edu]

This week, we meet as usual on Thursday, April 18 at 2:50 pm in WH 100E.
There will be a dinner afterwards organized by the speaker's host, Jenya
Sapir.

Speaker: Elizabeth Field (UIUC)

Title: Trees, dendrites, and the Cannon-Thurston map

Abstract: When $1\to H\to G\to Q\to 1$ is a short exact sequence of three
word-hyperbolic groups, Mahan Mj has shown that the inclusion map from $H$
to $G$ extends continuously to a map between the Gromov boundaries of $H$
and $G$. This boundary map is known as the Cannon-Thurston map. In this
context, Mitra associates to every point $z$ in the Gromov boundary of $Q$
an ``ending lamination'' on $H$ which consists of pairs of distinct points
in the boundary of $H$. We prove that for each such $z$, the quotient of
the Gromov boundary of $H$ by the equivalence relation generated by this
ending lamination is a dendrite, that is, a tree-like topological space.
This result generalizes the work of Kapovich-Lustig and
Dowdall-Kapovich-Taylor, who prove that in the case where $H$ is a free
group and $Q$ is a convex cocompact purely atoroidal subgroup of
$Out(F_N)$, one can identify the resultant quotient space with a certain
$R$-tree in the boundary of Culler-Vogtmann's Outer space.