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

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 , by running a random walk on generating -tuples of . 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:
http://dx.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