A ‘transversal’ for minimal invariant sets in the boundary of a CAT(0) group

Abstract

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>acting geometrically on a CAT(0) space<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding="application/x-tex">X</mml:annotation></mml:semantics></mml:math></inline-formula>we show there is a flat<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F subset-of upper X"><mml:semantics><mml:mrow><mml:mi>F</mml:mi><mml:mo>⊂</mml:mo><mml:mi>X</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">F\subset X</mml:annotation></mml:semantics></mml:math></inline-formula>of maximal dimension (denote it by<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"><mml:semantics><mml:mi>d</mml:mi><mml:annotation encoding="application/x-tex">d</mml:annotation></mml:semantics></mml:math></inline-formula>), whose boundary sphere intersects every minimal<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>-invariant subset of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript normal infinity Baseline upper X"><mml:semantics><mml:mrow><mml:msub><mml:mi mathvariant="normal">∂</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msub><mml:mi>X</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\partial _\infty X</mml:annotation></mml:semantics></mml:math></inline-formula>. As applications we obtain an improved dimension-dependent bound<disp-formula content-type="math/mathml">\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d i a m partial-differential Subscript Sub Subscript normal upper T Baseline upper X less-than-or-equal-to 2 pi minus arc cosine left-parenthesis minus StartFraction 1 Over d plus 1 EndFraction right-parenthesis"><mml:semantics><mml:mrow><mml:mi>diam</mml:mi><mml:mo>⁡</mml:mo><mml:msub><mml:mi mathvariant="normal">∂</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi/><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">T</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mi>X</mml:mi><mml:mo>≤</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo>−</mml:mo><mml:mi>arccos</mml:mi><mml:mo>⁡</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">\operatorname {diam}\partial _{_\mathrm {T}} X\leq 2\pi -\arccos \left (-\frac {1}{d+1}\right )</mml:annotation></mml:semantics></mml:math>\]</disp-formula>on the Tits-diameter of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential upper X"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">∂</mml:mi><mml:mi>X</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\partial X</mml:annotation></mml:semantics></mml:math></inline-formula>for non-rank-one groups, a necessary and sufficient dynamical condition for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>to be virtually Abelian, and we formulate a new approach to Ballmann’s rank rigidity conjectures.