Volume 15, issue 1 (2011)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Biharmonic functions on groups and limit theorems for quasimorphisms along random walks

Michael Björklund and Tobias Hartnick

Geometry & Topology 15 (2011) 123–143
Abstract

We show for very general classes of measures on locally compact second countable groups that every Borel measurable quasimorphism is at bounded distance from a quasi-biharmonic one. This allows us to deduce nondegenerate central limit theorems and laws of the iterated logarithm for such quasimorphisms along regular random walks on topological groups using classical martingale limit theorems of Billingsley and Stout. For quasi-biharmonic quasimorphisms on countable groups we also obtain integral representations using martingale convergence.

Keywords
quasimorphism, central limit theorem, biharmonic function, random walks, bounded cohomology
Mathematical Subject Classification 2000
Primary: 20P05, 60F05
References
Publication
Received: 29 April 2010
Revised: 3 November 2010
Accepted: 24 August 2010
Published: 25 January 2011
Proposed: Danny Calegari
Seconded: Leonid Polterovich, Benson Farb
Authors
Michael Björklund
Departement Mathematik
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland
Tobias Hartnick
Mathematics Department
Technion - Israel Institute of Technology
32000 Haifa
Israel