# Random subgroups of Thompson’s group $F$

### Sean Cleary

The City College of CUNY, New York, USA### Murray Elder

The University of Queensland, Brisbane, Australia### Andrew Rechnitzer

The University of British Columbia, Vancouver, Canada### Jennifer Taback

Bowdoin College, Brunswick, USA

## Abstract

We consider random subgroups of Thompson’s group F with respect to two natural stratifications of the set of all k-generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of *persistent* subgroups, whose isomorphism classes occur with positive density within the set of k-generator subgroups, for all sufficiently large k. Additionally, Thompson’s group provides the first example of a group without a generic isomorphism class of subgroup. Elements of F are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite (short for differentiably finite) and not algebraic. We then use the asymptotic growth to prove our density results.

## Cite this article

Sean Cleary, Murray Elder, Andrew Rechnitzer, Jennifer Taback, Random subgroups of Thompson’s group $F$. Groups Geom. Dyn. 4 (2010), no. 1, pp. 91–126

DOI 10.4171/GGD/76