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.

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