Combinatorics of tight geodesics and stable lengths

Abstract

We give an algorithm to compute the stable lengths of pseudo-Anosovs on the curve graph, answering a question of Bowditch. We also give a procedure to compute all invariant tight geodesic axes of pseudo-Anosovs. Along the way we show that there are constants <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than a 1 greater-than a 2"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">1&gt;a_1&gt;a_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the minimal upper bound on ‘slices’ of tight geodesics is bounded below and above by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a 1 Superscript xi left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ξ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">a_1^{\xi (S)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a 2 Superscript xi left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ξ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">a_2^{\xi (S)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ξ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\xi (S)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the complexity of the surface. As a consequence, we give the first computable bounds on the asymptotic dimension of curve graphs and mapping class groups. Our techniques involve a generalization of Masur–Minsky’s tight geodesics and a new class of paths on which their tightening procedure works.