## Notes of Panos Papasoglu’s Cambridge lecture 13-01-2017

Cut-points in ${CAT(0)}$ groups

Joint with Eric Swenson.

1. Introduction

We want to characterize group splittings geometrically. The prototype is Bowditch theroem: if ${G}$ is one-ended hyperbolic, not virtually a surface group. Then ${G}$ split over a 2-ended group iff there is a pair of points that separate its ideal boundary. (Excluding virtually surface groups is necessary since triangle groups have a circle boundary but do not split).

Theorem 1 (Papasoglu-Swenson 2009) Same is true for ${CAT(0)}$ groups.

How about splitting over ${{\mathbb Z}^n}$ ? If ${G}$ is ${CAT(0)}$ and splits over ${{\mathbb Z}^n}$, the visual boundary contains a separating round sphere.

This is known for Hadamard manifold groups, thanks to results of Schröder and Dunwoody-Swenson’s algebraic torus theorem.

Question. What if something smaller than a sphere or a circle separates the boundary ?

Theorem 2 Let ${G}$ be ${CAT(0)}$ and 1-ended, not a virtually surface group. Suppose ${n>2}$ points separate the visual boundary. Then ${G}$ splits over a 2-ended group.

2. ${CAT(0)}$ geometry

The visual boundary has an angle metric: supremum of angles under which a pair of visual points are seen. The Tits metric ${d_T}$ is the path metric that comes from the angle metric. Isometries act isometrically on the Tits metric. Although Tits metric may be infinite, it is useful, since Tits=${\infty}$ reflects hyperbolic behaviour, which is favorable. Difficulty: isometries act trivially on boundaries of stabilized flats.

Here is a result that gives some of a North-South dynamics.

Theorem 3 (${\pi}$-convergence) Let ${g_n\in G}$ be such that ${g_n x\rightarrow p\in\partial X}$ and ${g_n^{-1} x\rightarrow q\not=p\in\partial X}$. Assume that ${d_T (q,a)\geq\pi}$, then ${g_n a\rightarrow p}$.

3. Cactus trees

To get a splitting; we need an action on a tree. We define the cactus tree of a continuum. Say a continuum is ${n}$-thick in ${n-1}$ points never disconnect it and there is an ${n}$-tuple which disconnect it (call such a set an ${n}$-cut).

Theorem 4 Let ${X}$ be an ${n}$-thick continuum. Then there is an ${{\mathbb R}}$-tree encoding all ${n}$-cuts.

Our central observation is that crossing ${n}$-cuts are arranged in “wheels”: there is a cyclic order on “half-cuts”.

4. Proof

We want to use Levitt’s theorem: non-nesting actions by homeomorphisms on ${{\mathbb R}}$-trees can be replaced with isometric actions. Unfortunately, the non-nesting assumption need not be satisfied. For instance, let ${G=F_2 \star_{\mathbb Z} H}$ act on its Bass-Serre tree. One can modify the action by replacing ${F_2}$ vertices with copies of ${Cay(F_2)}$, leading to a non-isometric nesting action.

So here is our strategy;

1. Show action on cactus tree ${T}$ does not fix a point.
2. Show ${G}$ is rank 1, i.e. the Tits diameter of ${X}$ is infinite.
3. If ${G}$ action on ${T}$ is nesting, pass to a non-nesting quotient tree to get a splitting.
4. Otherwise, apply Bass-Serre-Rips.

1. We show that if ${G}$ fixes an ${n}$-cut, then it fixes a point. An ingredient is the fact that the Tits radius of an ${n}$-cut is at least ${\pi}$, combined with ${\pi}$-convergence. If there is a fixed point, a theorem by Ruane allows to conclude that ${G=H\times {\mathbb Z}}$ virtually. Similar arguments when ${G}$ stabilizes a wheel.

4. Nesting case.

Theorem 5 Suppose ${G}$ acts minimally and nestingly on an ${{\mathbb R}}$-tree ${T}$ with nonoverlapping translation intervals. Then ${G}$ splits over a stabilizer of an end of a cross-component.