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Statistical properties of subgroups of free groups

Statistical properties of subgroups of free groups
Statistical properties of subgroups of free groups
The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.
subgroups of free groups, finite group presentations, statistical properties, stallings graphs, partial injections, malnormality
Bassino, Frederique
68088a38-68ca-48a5-a60b-31ef08b893d6
Martino, Armando
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Nicaud, Cyril
5ab96cbd-893a-47ab-960c-9459bab3ff1a
Ventura, Enric
543ad8f8-4af2-41c5-91ec-7781d79bf647
Weil, Pascal
293bcff5-8465-42fc-8394-6f7d2c7431ef
Bassino, Frederique
68088a38-68ca-48a5-a60b-31ef08b893d6
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Nicaud, Cyril
5ab96cbd-893a-47ab-960c-9459bab3ff1a
Ventura, Enric
543ad8f8-4af2-41c5-91ec-7781d79bf647
Weil, Pascal
293bcff5-8465-42fc-8394-6f7d2c7431ef

Bassino, Frederique, Martino, Armando, Nicaud, Cyril, Ventura, Enric and Weil, Pascal (2010) Statistical properties of subgroups of free groups. Mathematics. (Submitted)

Record type: Article

Abstract

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.

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Submitted date: 25 January 2010
Keywords: subgroups of free groups, finite group presentations, statistical properties, stallings graphs, partial injections, malnormality
Organisations: University of Southampton

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Local EPrints ID: 142343
URI: https://eprints.soton.ac.uk/id/eprint/142343
PURE UUID: 87fe90a1-4ca3-4e7f-8359-b24b2ceba3c1

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Date deposited: 31 Mar 2010 14:29
Last modified: 25 Feb 2019 17:32

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Contributors

Author: Frederique Bassino
Author: Armando Martino
Author: Cyril Nicaud
Author: Enric Ventura
Author: Pascal Weil

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