Implementation of the Conjugacy Solution in Thompson's Group F

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

2. Group Conjugacy • Conjugacy: In a group G, the elements g1,g2ϵGare conjugate if there exists hϵGsuch that g1=hg2h-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

3. Thompson’s Group F • 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 : xo & x1

4. Tree & Strand Diagrams • Tree diagram: Convert breakpoints into trees Domain x0 Codomain

5. 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 x0 Codomain

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

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

8. ASD for x1 Annular Strand Diagrams (ASD) • 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 a split a merge A free loop

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

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

11. 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 reducedASDs 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. A1 and A2 are isotopic

12. 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

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

14. 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 reducedASDs 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 Aand B can be encoded into two planar graphs PAand PB respectively such that A and B are isotopic if and only if PAand PB are isomorphic. • Hence isotopy detection reduced to isomorphism of planar graphs

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

16. 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<Vertex>) • Method: close() • Annular • vertices (LinkedList<Vertex>) • cuttingPath(LinkedList<Edge>): a sequence of edges in the cutting path • stackReduceSplits (Stack<Vertex>): stores split vertices to identify reductions • Methods: reduce(), getComponents(), encodeToPlanarGraph()

17. Easy to see: algorithm is linear up to ASD creation

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

19. 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 : e1 = e4 • For reduction II : e1 = e4 or e2 = e5 • 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 stackReduceSplitsduring each reduction I or II

20. Therefore algorithm is linear up to Reduce

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

22. 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)

23. Linear up to CC Labeling

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

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

26. Encoding to Planar Graph • Recall: isotopy of ASDs was reduced to isomorphism of planar graphs • Any two connected, reduced ASDs Aand B can be encoded into two planar graphs PAand PB respectively such that A and B are isotopic if and only if PAand PB 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, ink) uniquely identifies the edge class of ek

27. Encoding to Planar Graph • Recall: isotopy of ASDs was reduced to isomorphism of planar graphs • Any two connected, reduced ASDs Aand B can be encoded into two planar graphs PAand PB respectively such that A and B are isotopic if and only if PAand PB 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, ink) uniquely identifies the edge class of ek

28. Linear up to Encoding to Planar Graph

29. 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

30. 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

31. Implementation

32. 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(n2) implementations • Our software can be extended to solve the conjugacy problems in the two other Thompson’s Groups: • Group V • Group T

33. 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 6th annular ACM symposium on Theory of Computing, 1974, pp.172-184 [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

34. Question Time !!

35. The Comprehensive Data Structure

36. Evaluation

37. Verification Nabil Hossain