1. 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.
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.
DATA STRUCTURE
All linked lists are doubly linked.
Node – a node in a linked list.
Annular strand diagram for x1 obtained by closing.
DEFINITIONS
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.
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.
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.
[N is the sum of the lengths of the input words.]
Annular Strand Diagram Generation: Constant number of vertices and edges per unit length of a word.
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.
Isomorphism Check: Uses the linear time algorithm for isomorphism problem in planar graphs proposed by Hopcroft and Wong [2].
Reductions – rules for simplifying ASDs
An ASD is reduced if it is not subject to any more reductions.
Example of reducing an ASD:
THOMPSON’S GROUP F
A group of piecewise-linear homeomorphisms of the interval [0,1]
such that:
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:
a merge a split
Strand diagrams for the generators x0 and x1, and their inverses:
Concatenation (composition of elements):
CUTTING PATH
A directed path from the inside of the annulus to the outside such that:
Updating the cutting path during a reduction move:
REFERENCES
[1] James Belk and Francesco Matucci, Conjugacy and dynamics in Thompson’s groups, Geometriae Dedicata. Available online at
[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
[3] Nabil Hossain, Algorithm for the Conjugacy Problem in Thompson’s Group F, 2013, Online; accessed 5 May, 2013
Cutting path can identify component ordering: