Sub-dominant Cogrowth Behavior and the Viability of Deciding Amenability Numerically

ABSTRACT

We critically analyze a recent numerical method due to the first author, Rechnitzer, and van Rensburg, which attempts to detect amenability in a finitely generated group by numerically estimating its asymptotic cogrowth rate. We identify two potential sources of error. We then propose a modification of the method that enables it to easily compute surprisingly accurate estimates for initial terms of the cogrowth sequence.

KEYWORDS:

• amenable group
• cogrowth function
• Følner function
• Kesten's criterion
• return probability
• Metropolis algorithm
• R. Thompson's group F

2010 AMS SUBJECT CLASSIFICATION:

• 20F69
• 20F65
• 05A15
• 60J20

Acknowledgments

The authors wish to thank Andrew Rechnitzer, Andrew Elvey-Price, and the anonymous reviewer for helpful feedback on this work.

Additional information

Funding

This research was supported by the Australian Research Council grant FT110100178.