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Extremality of the Rotation Quasimorphism on the Modular Group


Louwsma, Joel Ryan (2011) Extremality of the Rotation Quasimorphism on the Modular Group. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0Y6J-VP31.


For any element A of the modular group PSL(2,Z), it follows from work of Bavard that scl(A) is greater than or equal to rot(A)/2, where scl denotes stable commutator length and rot denotes the rotation quasimorphism. Sometimes this bound is sharp, and sometimes it is not. We study for which elements A in PSL(2,Z) the rotation quasimorphism is extremal in the sense that scl(A)=rot(A)/2. First, we explain how to compute stable commutator length in the modular group, which allows us to experimentally determine whether the rotation quasimorphism is extremal for a given A. Then we describe some experimental results based on data from these computations.

Our main theorem is the following: for any element of the modular group, the product of this element with a sufficiently large power of a parabolic element is an element for which the rotation quasimorphism is extremal. We prove this theorem using a geometric approach. It follows from work of Calegari that the rotation quasimorphism is extremal for a hyperbolic element of the modular group if and only if the corresponding geodesic on the modular surface virtually bounds an immersed surface. We explicitly construct immersed orbifolds in the modular surface, thereby verifying this geometric condition for appropriate geodesics. Our results generalize to the 3-strand braid group and to arbitrary Hecke triangle groups.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:stable commutator length; quasimorphism; Bavard duality; modular group; rotation quasimorphism
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Calegari, Danny C.
Thesis Committee:
  • Calegari, Danny C. (chair)
  • Aschbacher, Michael
  • Day, Matthew B.
  • Marcolli, Matilde
Defense Date:26 May 2010
Record Number:CaltechTHESIS:05132010-155930760
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5807
Deposited By: Joel Louwsma
Deposited On:17 May 2011 23:13
Last Modified:28 Oct 2021 19:12

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