Proceedings from Top Secret Topology Seminar.

Disclaimer: Handle with care. Several unproved (possibly untrue) statements lie ahead.

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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

1. Order Isomorphic
   
2. Quotient Topology
3. 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.

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[Bowb] B.H. Bowditch, Treelike structures arising from continua and convergence groups, Memoirs Amer. Math. Soc. , Vol 139, Number 662, May 1999.