Proceedings from Top Secret Topology Seminar.
Disclaimer: Handle with care. Several unproved (possibly untrue) statements lie ahead.
Le G be a word-hyperbolic group and \( \partial G \) its boundary.
Theorem (Bestvina, Messa, 1991)
If G has one end, then \( \partial G \) is connected and locally connected.
The second part (locally connected), was proved under the assumption that \( \partial G \) contains no cut points. The theory of \( \mathbb{R} \)treeswas used to establish:
Theorem (Bowditch, Swarup)
If G has one end, then \( \partial G \) contains no cut points.
Proof:
Step 1: Assume that \( \partial G \) has a cut point. We will show that G is more than one-ended leading to proof by contradiction.
Basic Notions 1
- Word Hyperbolic Group
- \( \partial G \)
- End of a Group
- Cut Point
- Locally Connected
\( \partial G \) is compact metric space
- G is word hyperbolic implies \( \tilde {G} = G \cup \partial G \) is a compact metric space and \( \partial G \) is a closed (hence compact) subset of \( \tilde {G} \). (Bridson, Chapter III.H , Proposition 3.7, Pg 429)
Sequence of ideas leading to the definition of Dendrite (from BowB)
Uniquely Arc Connected Space
A uniquely arc-connected space, T, is a hausdorff topological space in which every pair of distinct points are joined by a unique arc (subset homeomorphic to a closed real interval).
Equivalently, it can be defined as a path connected Hausdorff space which contains no topologically embedded circle.
Real Tree
A real tree is a uniquely arc connected space T such that for every \( x \in T \) and every neighborhood V of x, there is a neighborhood U of x such that if \( y \in U \) then \( [x, y] \subseteq V \)
Topologically Dense
Suppose \( R \subseteq T \). We use the term topologically dense to mean that every open subset of T contains a point of R.
Separable Real Tree
A real tree T is separable if it contains a countable dense subset.
Dendron
A dendron is a compact real tree
Dendrite
A dendrite is a separable dendron.
Hence a dendrite is a uniquely arc connected, hausdorff, topological space which is compact and has a countable dense subset (hence is separable).
A dendrite is metrizable, locally connected (theorem).
Yet another definition of dendrite: a compact separable \( \mathbb{R} \) Tree.
We want to make \(\partial G \) into a dendrite.
Claim: G acts naturally and minimally on \( \partial G \) (recall G is word – hyperbolic).
Minimal Group Action
The action of G on X is minimal if Gx is dense in X for each \( x \in X \). This is equivalent to saying that X has no proper G – invariant closed subset of X. (why is it equivalent?)
Hence if there is one cut point in \( \partial G \) then there exists many cut points in \( \partial G \).
Define an equivalence relation \( \sim \) on \( \partial G \). Two points \( x, y \in M \) are not equivalent if there is a collection C of cut points in \( \partial G \) that each separate x from y and which is order isomorphic to the rationals.
Next assertion follows from a Theorem by Bowditch (Bowb, Theorem 5.2, Pg. 52)
Assertion: \( \partial G / \sim \) is a dendrite D.
Picture to keep in mind
Claim: \( \partial G / \sim \) is not a point
This follows from: (Bowb, Theorem 6.1, Pg. 52)
We will be considering the action of G on D \ {endpoints}, which is an \( \mathbb{R} \) tree.
Basic Notions 2
- Order Isomorphic
- Quotient Topology
- End points of a Dendrite
Summary:
- G is an one ended hyperbolic group, \( \partial G \) is it’s Gromov Boundary.
- Assume \( \partial G \) has a cut point x .
- Since natural action of G on its boundary is minimal, hence Gx is dense in \( \partial G \); that is there are many cut points.
- Quotient out the points in \( \partial G \) which are not separated by an order isomorphic set of cut points.
- The quotient space is a dendrite D (compact, uniquely arc connected, separable, metric space)
- D is not trivial.
- Remove the endpoints of D to get a real tree T.
Coming up next
We will be analyzing the action of G on this real tree to find that G splits over a finite group.
[Bowb] B.H. Bowditch, Treelike structures arising from continua and convergence groups, Memoirs Amer. Math. Soc. , Vol 139, Number 662, May 1999.
