# Groups, Geometry, and Dynamics

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

**DOI: 10.4171/GGD/76**

Random subgroups of Thompson’s group `F`

^{[1]}, Murray Elder

^{[2]}, Andrew Rechnitzer

^{[3]}and Jennifer Taback

^{[4]}(1) Department of Mathematics, The City College of CUNY, Convent Avenue and 138th Street, NY 10031, NEW YORK, UNITED STATES

(2) Mathematics, The University of Queensland, St Lucia, QLD 4072, BRISBANE, AUSTRALIA

(3) Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, BC V6T 1Z2, VANCOUVER, CANADA

(4) Department of Mathematics, Bowdoin College, 8600 College Station, ME 04011, BRUNSWICK, UNITED STATES

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 S, Elder M, Rechnitzer A, Taback J. Random subgroups of Thompson’s group `F`. *Groups Geom. Dyn.* 4 (2010), 91-126. doi: 10.4171/GGD/76