Cut-points in groups
Joint with Eric Swenson.
We want to characterize group splittings geometrically. The prototype is Bowditch theroem: if is one-ended hyperbolic, not virtually a surface group. Then 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 groups.
How about splitting over ? If is and splits over , 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 be and 1-ended, not a virtually surface group. Suppose points separate the visual boundary. Then splits over a 2-ended group.
The visual boundary has an angle metric: supremum of angles under which a pair of visual points are seen. The Tits metric 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= 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 (-convergence) Let be such that and . Assume that , then .
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 -thick in points never disconnect it and there is an -tuple which disconnect it (call such a set an -cut).
Theorem 4 Let be an -thick continuum. Then there is an -tree encoding all -cuts.
Our central observation is that crossing -cuts are arranged in “wheels”: there is a cyclic order on “half-cuts”.
We want to use Levitt’s theorem: non-nesting actions by homeomorphisms on -trees can be replaced with isometric actions. Unfortunately, the non-nesting assumption need not be satisfied. For instance, let act on its Bass-Serre tree. One can modify the action by replacing vertices with copies of , leading to a non-isometric nesting action.
So here is our strategy;
- Show action on cactus tree does not fix a point.
- Show is rank 1, i.e. the Tits diameter of is infinite.
- If action on is nesting, pass to a non-nesting quotient tree to get a splitting.
- Otherwise, apply Bass-Serre-Rips.
1. We show that if fixes an -cut, then it fixes a point. An ingredient is the fact that the Tits radius of an -cut is at least , combined with -convergence. If there is a fixed point, a theorem by Ruane allows to conclude that virtually. Similar arguments when stabilizes a wheel.
4. Nesting case.
Theorem 5 Suppose acts minimally and nestingly on an -tree with nonoverlapping translation intervals. Then splits over a stabilizer of an end of a cross-component.