53 research outputs found
Some geodesic problems in groups
We consider several algorithmic problems concerning geodesics in finitely
generated groups. We show that the three geodesic problems considered by
Miasnikov et al [arXiv:0807.1032] are polynomial-time reducible to each other.
We study two new geodesic problems which arise in a previous paper of the
authors and Fusy [arXiv:0902.0202] .Comment: 6 page
Two Non-holonomic Lattice Walks in the Quarter Plane
We present two classes of random walks restricted to the quarter plane whose
generating function is not holonomic. The non-holonomy is established using the
iterated kernel method, a recent variant of the kernel method. This adds
evidence to a recent conjecture on combinatorial properties of walks with
holonomic generating functions. The method also yields an asymptotic expression
for the number of walks of length n
Random subgroups of Thompson's group
We consider random subgroups of Thompson's group with respect to two
natural stratifications of the set of all 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 {\em persistent} subgroups, whose
isomorphism classes occur with positive density within the set of -generator
subgroups, for all sufficiently large . Additionally, Thompson's group
provides the first example of a group without a generic isomorphism class of
subgroup. Elements of 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 and not algebraic.
We then use the asymptotic growth to prove our density results.Comment: 37 pages, 11 figure
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