We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g ≥ 1 2 cosh(r) where r denotes the radius of any isometrically embedded ball in M. Assuming an unpublished result of Pitts and Rubinstein improves this to g ≥ 1 2 cosh(r) + 1 2. We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls.
Bachman, David 1 ; Cooper, Daryl 2 ; White, Matthew E 1
@article{10_2140_agt_2004_4_31,
author = {Bachman, David and Cooper, Daryl and White, Matthew E},
title = {Large embedded balls and {Heegaard} genus in negative curvature},
journal = {Algebraic and Geometric Topology},
pages = {31--47},
year = {2004},
volume = {4},
number = {1},
doi = {10.2140/agt.2004.4.31},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.31/}
}
TY - JOUR AU - Bachman, David AU - Cooper, Daryl AU - White, Matthew E TI - Large embedded balls and Heegaard genus in negative curvature JO - Algebraic and Geometric Topology PY - 2004 SP - 31 EP - 47 VL - 4 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.31/ DO - 10.2140/agt.2004.4.31 ID - 10_2140_agt_2004_4_31 ER -
%0 Journal Article %A Bachman, David %A Cooper, Daryl %A White, Matthew E %T Large embedded balls and Heegaard genus in negative curvature %J Algebraic and Geometric Topology %D 2004 %P 31-47 %V 4 %N 1 %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.31/ %R 10.2140/agt.2004.4.31 %F 10_2140_agt_2004_4_31
Bachman, David; Cooper, Daryl; White, Matthew E. Large embedded balls and Heegaard genus in negative curvature. Algebraic and Geometric Topology, Tome 4 (2004) no. 1, pp. 31-47. doi: 10.2140/agt.2004.4.31
[1] , , Heegaard genus of closed orientable Seifert 3–manifolds, Invent. Math. 76 (1984) 455
[2] , A covering theorem for hyperbolic 3–manifolds and its applications, Topology 35 (1996) 751
[3] , , On limits of tame hyperbolic 3–manifolds, J. Differential Geom. 43 (1996) 1
[4] , , The min-max construction of minimal surfaces, from: "Surveys in differential geometry, Vol. VIII (Boston, MA, 2002)", Surv. Differ. Geom., VIII, Int. Press, Somerville, MA (2003) 75
[5] , The volume of a closed hyperbolic 3–manifold is bounded by $\pi$ times the length of any presentation of its fundamental group, Proc. Amer. Math. Soc. 127 (1999) 941
[6] , On triangulations of surfaces, Topology Appl. 40 (1991) 189
[7] , , Nielsen methods and groups acting on hyperbolic spaces, Geom. Dedicata 98 (2003) 95
[8] , The volume of hyperbolic alternating link complements, Proc. London Math. Soc. $(3)$ 88 (2004) 204
[9] , , Applications of minimax to minimal surfaces and the topology of 3–manifolds, from: "Miniconference on geometry and partial differential equations, 2 (Canberra, 1986)", Proc. Centre Math. Anal. Austral. Nat. Univ. 12, Austral. Nat. Univ. (1987) 137
[10] , The isoperimetric and Willmore problems, from: "Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000)", Contemp. Math. 288, Amer. Math. Soc. (2001) 149
[11] , The isoperimetric problem, from: "Global theory of minimal surfaces", Clay Math. Proc. 2, Amer. Math. Soc. (2005) 175
[12] , Minimal surfaces in geometric 3–manifolds, from: "Global theory of minimal surfaces", Clay Math. Proc. 2, Amer. Math. Soc. (2005) 725
[13] , Genus 2 closed hyperbolic 3–manifolds of arbitrarily large volume, from: "Proceedings of the Spring Topology and Dynamical Systems Conference (Morelia City, 2001)" (2001/02) 317
[14] , Injectivity radius and fundamental groups of hyperbolic 3–manifolds, Comm. Anal. Geom. 10 (2002) 377
[15] , A diameter bound for closed, hyperbolic 3–manifolds
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