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The product replacement algorithm and Kazhdan's property (T)


Authors: Alexander Lubotzky and Igor Pak
Journal: J. Amer. Math. Soc. 14 (2001), 347-363
MSC (2000): Primary 60B15; Secondary 05C25, 22D10, 60J10
DOI: https://doi.org/10.1090/S0894-0347-00-00356-8
Published electronically: October 18, 2000
MathSciNet review: 1815215
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Abstract: The ``product replacement algorithm'' is a commonly used heuristic to generate random group elements in a finite group $G$, by running a random walk on generating $k$-tuples of $G$. While experiments showed outstanding performance, the theoretical explanation remained mysterious. In this paper we propose a new approach to the study of the algorithm, by using Kazhdan's property (T) from representation theory of Lie groups.


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Additional Information

Alexander Lubotzky
Affiliation: Department of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel
Email: alexlub@math.huji.ac.il

Igor Pak
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

DOI: https://doi.org/10.1090/S0894-0347-00-00356-8
Keywords: Random walks on groups, Kazhdan's property (T), nilpotent groups
Received by editor(s): January 5, 2000
Received by editor(s) in revised form: August 23, 2000
Published electronically: October 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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