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# Cut points in Boundary – 1

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

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.

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

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