Implementation of a Solution to the
Conjugacy Problem in Thompson’s Group F
James Belk, Nabil Hossain, Francesco Matucci, and Robert W. McGrail
ABSTRACT
We present an efficient implementation of the solution to the
conjugacy problem in Thompson's group F, a certain infinite group
whose elements are piecewise-linear homeomorphisms of the unit
interval [0,1]. This algorithm checks for conjugacy by constructing and
comparing directed graphs called strand diagrams. We provide a
description of our solution algorithm, including the data structure that
represents strand diagrams and supports manipulations of strand
diagrams. The algorithm theoretically achieves an O(n) bound in the
size of the input, and we provide a O(n2) working solution.
DATA STRUCTURE
ANNULAR STRAND DIAGRAMS
• Finite directed graphs embedded in the annulus
obtained by closing a strand diagram.
• Every directed cycle winds counterclockwise around the
central hole.
• All linked lists are doubly linked.
• Node – a node in a linked list.
Annular strand diagram for x1 obtained by closing.
DEFINITIONS
Reductions – rules for simplifying ASDs
Conjugacy: In a group G, two elements g1 and g2 are conjugate if
there exists h ϵ G such that g1 = hg2h-1.
Conjugacy Problem: In a group G with a given generating set S, the
conjugacy problem is the decision problem of determining whether
two given words w1 and w2 in S are conjugate.
O(N) ALGORITHM
Theorem (Belk and Matucci [1]): Let g and h be elements of F. Let
ga and ha be their reduced annular strand diagrams respectively. Then
g and h are conjugate if ga and ha are isotopic.
• An ASD is reduced if it is not subject to any more reductions.
Example of reducing an ASD:
Isotopic Directed Graphs: Two directed graphs embedded on a
surface are isotopic if one is the image of the other under some
continuous deformation of the surface.
THOMPSON’S GROUP F
[N is the sum of the lengths of the input words.]
 A group of piecewise-linear homeomorphisms of the interval [0,1]
such that:
Annular Strand Diagram Generation: Constant number of vertices
and edges per unit length of a word.
1) For any element, each piece has a slope that is a power of 2; and
2) Breakpoints between pieces have dyadic rational coordinates.
 Elements of F can be represented in finite acyclic directed graphs
called strand diagrams [1] which:
 are embedded on the unit square;
 have a source, and a sink; and
 any other vertex is either a merge or a split:
CUTTING PATH
A directed path from the inside of the annulus to the
outside such that:
Reduce: Uses a stack, storing splits at the points of concatenations.
Pop splits from the stack and check for reductions. Perform possible
reductions, and add neighboring splits to stack. When stack is empty,
the ASD is reduced.
Connected Component Labeling: Breadth first search along edges
in cutting path.
Encoding to Planar Graph: Creates a planar graph having O(V)
vertices and O(E) edges, where V and E are the vertices and edges
respectively in the original ASD for the element.
a merge
a split
Updating the cutting path during a reduction move:
Isomorphism Check: Uses the linear time algorithm for isomorphism
problem in planar graphs proposed by Hopcroft and Wong [2].
Strand diagrams for the generators x0 and x1, and their inverses:
REFERENCES
Cutting path can identify component ordering:
[1] James Belk and Francesco Matucci, Conjugacy and dynamics in Thompson’s groups, Geometriae
Dedicata. Available online at http://link.springer.com/article/10.1007/s10711-013-9853-2.
[2] John Hopcroft and Jin-Kue Wong, Linear Time Algorithm for Isomorphism of Planar
Graphs,Proceedings of the 6th annular ACM symposium on Theory of Computing, 1974, pp.172-184
Concatenation (composition of elements):
[3] Nabil Hossain, Algorithm for the Conjugacy Problem in Thompson’s Group F, 2013,
http://www.asclab.org/asc/nhossain/conjugacyF. Online; accessed 5 May, 2013