# Groups, Geometry, and Dynamics

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**Volume 4, Issue 1, 2010, pp. 91–126**

**DOI: 10.4171/GGD/76**

Published online: 2009-12-23

Random subgroups of Thompson’s group `F`

^{[1]}, Murray Elder

^{[2]}, Andrew Rechnitzer

^{[3]}and Jennifer Taback

^{[4]}(1) The City College of CUNY, New York, USA

(2) The University of Queensland, Brisbane, Australia

(3) The University of British Columbia, Vancouver, Canada

(4) Bowdoin College, Brunswick, USA

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.

*Keywords: *Thompson's group `F`, asymptotic density, subgroup spectrum, visible subgroup, persistent subgroup, statistical group theory, asymptotic group theory, D-finite generating function, non-algebraic generating function

Cleary Sean, Elder Murray, Rechnitzer Andrew, Taback Jennifer: Random subgroups of Thompson’s group `F`. *Groups Geom. Dyn.* 4 (2010), 91-126. doi: 10.4171/GGD/76