Senior Project Board Implementation of the Solution to the Conjugacy Problem in Thompson’s Group F by Nabil Hossain Advisers: James Belk & Robert McGrail Division of Science, Mathematics & Computing Bard College

Conjugacy: In a group G, the elements g 1, g 2 ϵ G are conjugate if there exists hϵG such that g 1 = hg 2 h -1 Word: Given a group G with a generating set S, a word in S is an arbitrary product of the generators in S and their inverses. – A word represents an element of G Conjugacy Problem: Find an algorithm to determine whether two words in a group represent conjugate elements Group Conjugacy

Infinite group of piecewise linear homeomorphisms of [0,1] Pieces have finitely many breakpoints satisfying the conditions: – Every slope is a power of 2 – Every breakpoint has dyadic rational coordinates rational numbers with denominators that are powers of 2 Generators : x o & x 1 Thompson’s Group F

Tree diagram: Convert breakpoints into trees Tree & Strand Diagrams Domain Codomain x0x0

Tree & Strand Diagrams Tree diagram: Convert breakpoints into trees Strand Diagram: – Glue domain and codomain trees at leaves – Edge directions: “Domain” vertices “Codomain” vertices Domain Codomain x0x0

Given a strand diagram S, we generate S -1 by: – reversing the edge directions in S, and then – flipping the resultant strand diagram vertically Strand Diagrams for Generators of F

Concatenation (composition of elements) Closing (produces the Annular Strand Diagram) Strand Diagram Manipulations

Finite digraphs embedded in the annulus such that – Every vertex is a split or a merge – Every directed cycle winds counterclockwise around the central hole Can have free loops: directed cycles without any vertices Annular Strand Diagrams (ASD) a split a merge ASD for x 1 A free loop

Reductions A reduction of an ASD simplifies the directed graph using one of the three moves shown below. An ASD is reduced if it is subject to no further reductions These moves ensure that each ASD reduces to a unique normal form

Red and blue subject to type II move Green subject to type I move Example of Reducing an ASD Key Points: – Reducing can change the number of components in an ASD – Components are concentric

Isotopy of ASDs & Conjugacy Theorem Two ASDs are isotopic if one can be obtained from the other by some continuous motion in the annulus. Isotopic ASDs are considered equal Conjugacy Theorem (Belk and Matucci): Let a and b be words representing elements of Thompson’s group F. Let A and B be the corresponding strand diagrams, and let A′ and B′ be the reduced ASDs obtained by closing A and B and then reducing. Then a and b represent conjugate elements of F if and only if A′ and B′ are isotopic. A 1 and A 2 are isotopic

The Cutting Path Path from inside hole of the annulus to the outside that crosses at least one edge in the ASD Required to keep track of order of components when reducing an ASD Every ASD produced by closing a strand diagram has a cutting path Allowed NOT Allowed

Example of Updating the Cutting Path Rules to update the cutting path during reduction moves: Example:

Solution to the Conjugacy Problem in F Recall Conjugacy Theorem – Let a and b be words representing elements of Thompson’s group F. Let A and B be the corresponding strand diagrams, and let A′ and B′ be the reduced ASDs obtained by closing A and B and then reducing. Then a and b represent conjugate elements if and only if A′ and B′ are isotopic. Theorem: Any two connected, reduced ASDs A and B can be encoded into two planar graphs P A and P B respectively such that A and B are isotopic if and only if P A and P B are isomorphic. Hence isotopy detection reduced to isomorphism of planar graphs

Algorithm for the Conjugacy Problem in F Let n = |w1|+|w2|

The Data Structure Edge – class (Tuple of integers): stores input-output type of the edge (i.e. left, right) – Methods: combineEdge(), makeFreeLoop() Vertex – 4 Edges: Left Parent, Right Parent, Left Child, Right Child – type: an element in the set {source, sink, merge, split}  uniquely identifies the valid Edge objects associated with the vertex Graph – stores planar graphs encoded from reduced ASD Strand source,sink : for fast concatenation, closing vertices (LinkedList ) Method: close() Annular – vertices (LinkedList ) – cuttingPath (LinkedList ): a sequence of edges in the cutting path – stackReduceSplits (Stack ): stores split vertices to identify reductions – Methods: reduce(), getComponents(), encodeToPlanarGraph()

Easy to see: algorithm is linear up to ASD creation

stackReduceSplits initially stores all splits at concatenations (exposed to reduction II ) while (!stackReduceSplits.isEmpty()): Vertex v = stackReduceSplits.pop() if reduction I possible at v : reduction I ( v ) else if reduction II possible at v : reduction II ( v ) Perform all reduction III (check for adjacent free loops in cuttingPath) – NOTE: reduction I or II can add vertices to stackReduceSplits reduce()

Analysis of reduce() Updating the cutting path takes O(1) per reduction Checking for free loops (+ creating) takes O(1) per reduction  For reduction I : e 1 = e 4  For reduction II : e 1 = e 4 or e 2 = e 5 Total number of reductions is O(n) – each reduction I or II removes two vertices Total check for reductions is O(n) – constant number of splits put into stackReduceSplits during each reduction I or II

Therefore algorithm is linear up to Reduce

Connected Component Labeling Breadth first search along vertices connected to edges in cutting path in concentric order

Connected Component Labeling Breadth first search along vertices connected to edges in cutting path in concentric order Analysis: Sum of vertices and edges in all the connected components is bounded by the sum of the vertices and edges in the reduced ASD – Hence CC Labeling is O(n)

Linear up to CC Labeling

Encoding to Planar Graph Recall: isotopy of ASDs was reduced to isomorphism of planar graphs – Any two connected, reduced ASDs A and B can be encoded into two planar graphs P A and P B respectively such that A and B are isotopic if and only if P A and P B are isomorphic.

Encoding to Planar Graph Recall: isotopy of ASDs was reduced to isomorphism of planar graphs – Any two connected, reduced ASDs A and B can be encoded into two planar graphs P A and P B respectively such that A and B are isotopic if and only if P A and P B are isomorphic. The encoding function uniquely encodes each edge class – There is a free loop class – There are 9 classes between merges and splits

Encoding to Planar Graph Recall: isotopy of ASDs was reduced to isomorphism of planar graphs – Any two connected, reduced ASDs A and B can be encoded into two planar graphs P A and P B respectively such that A and B are isotopic if and only if P A and P B are isomorphic. The encoding function uniquely encodes each edge class – There is a free loop class – There are 9 classes between merges and splits The number of edges (u, in k ) uniquely identifies the edge class of e k

Encoding to Planar Graph Recall: isotopy of ASDs was reduced to isomorphism of planar graphs – Any two connected, reduced ASDs A and B can be encoded into two planar graphs P A and P B respectively such that A and B are isotopic if and only if P A and P B are isomorphic. The encoding function uniquely encodes each edge class – There is a free loop class – There are 9 classes between merges and splits. Analysis: Given a CC with |v| vertices and |e| edges, its encoded planar graphs has O(|v|) vertices and O(|e|) edges – Hence encoding to planar graph is O(n) The number of edges (u, in k ) uniquely identifies the edge class of e k

Linear up to Encoding to Planar Graph

Isomorphism Check Uses O(|V|) algorithm proposed by Hopcroft and Wong (1974) for the isomorphism problem in planar graphs – Their algorithm accepts multigraphs For each pair of corresponding planar graphs, check whether the graphs are isomorphic. – If all such pairs are isomorphic, then the corresponding elements are conjugate

Isomorphism Check Uses O(|V|) algorithm proposed by Hopcroft and Wong (1974) for the isomorphism problem in planar graphs – Their algorithm accepts multigraphs For each pair of corresponding planar graphs, check whether the graphs are isomorphic. – If all such pairs are isomorphic, then the corresponding elements are conjugate This isomorphism checker is mainly theoretical and has not been implemented yet

Implementation

Conclusion Contributions: – a theoretical O(n) algorithm for conjugacy problem in Thompson’s Group F using directed graphs called annular strand diagrams – a data structure for storing and manipulating ASDs – O(n 2 ) implementations Our software can be extended to solve the conjugacy problems in the two other Thompson’s Groups:  Group V  Group T

References [1] James Belk and Francesco Matucci, Conjugacy and Dynamics in Thompson’s Groups, preprint (2013). [2] John Hopcroft and Jin-Kue Wong, Linear Time Algorithm for Isomorphism of Planar Graphs,Proceedings of the 6 th 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, 2013http://

Question Time !!

The Comprehensive Data Structure

Evaluation

Verification 6/10/2016Nabil Hossain37