2015/07/31 by Christopher K. Atkinson, Christopher Atkinson, David Futer
Mathematics · #Bounded function #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Hyperbolic manifold #Manifold (fluid mechanics) #Orbifold #Torsion (gastropod) #Volume (thermodynamics) #math.GT #msc:57M50 #msc:57M60 #msc:57R18
paper · pdf · doi:10.1090/tran/6920
published as Trans. Amer. Math. Soc. 369 (2017), Issue 8, 5809-5827 · 17 pages, 1 figure. v2 contains minor edits. To appear in Transactions of the AMS
arxiv created 2016/02/18 · openalex publication_date 2016/02/29 · openalex created_date 2016/06/24 · arxiv updated 2017/05/09 · openalex updated_date 2026/08/06
For each natural number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n ≥ 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we determine the unique lowest volume hyperbolic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> –orbifold whose torsion orders are bounded below by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This lowest volume orbifold has base space the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> –sphere and singular locus the figure– <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="8"> <mml:semantics> <mml:mn>8</mml:mn> <mml:annotation encoding="application/x-tex">8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> knot, marked <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We apply this result to give sharp lower bounds on the volume of a hyperbolic manifold in terms of the order of elements in its symmetry group.