Algorithms for groups of homeomorphisms Susan Hermiller Joint work with Collin Bleak, Tara Brough
Overview • The groups PL + ( I ) and F • Algorithmic questions for PL + ( I ) and F • Solvability of finitely generated subgroups of PL + ( I ) • More on algorithms
The group PL + ( I ) PL + ( I ) := the group of Piecewise Linear orientation-preserving homeomorphisms : [0 , 1] → [0 , 1]. 1 0 0 1 Note. PL + ( I ) is not finitely generated. Instead consider any finitely generated (or finitely presented) sub- group of PL + ( I ).
Thompson’s group F R. Thompson’s group F := subgroup of PL homeomorphisms with all slopes in { 2 n | n ∈ Z } and breakpoints in Z [ 1 2 ]. F = � x 0 , x 1 | [ x 0 x − 1 1 , x − 1 0 x 1 x 0 ] , [ x 0 x − 1 1 , x − 2 0 x 1 x 2 0 ] � is finitely presented. if x ∈ [0 , 1 x 2 ] if x ∈ [0 , 1 2 x 4 ] 2 x − 1 if x ∈ [ 1 2 , 5 8 ] x + 1 if x ∈ [ 1 4 , 1 2 x 0 = x 1 = 2 ] x + 1 if x ∈ [ 5 8 , 3 4 4 ] 2 x + 1 1 if x ∈ [ 1 8 2 , 1] 1 2 x + 1 if x ∈ [ 3 2 4 , 1] 2 3/4 x x o 1 1/2 1/2 1/2 1/4 1/2
The groups PL + ( I ) and F Rmk. There are many ways to represent an element g of R. Thompson’s group F : • g = a word over { x ± 1 0 , x ± 1 1 } • g is determined by its breakpoints and slopes • g is a pair of rooted binary trees with equal numbers of leaves. • etc. ...
Algorithmic questions for PL + ( I ) and F , I: The Word Problem Fix a set S = { f 1 , ..., f m } of elements of PL + ( I ). The Word Problem (WP) for the subgroup � S � of PL + ( I ) asks: Q. Is there an algorithm that, upon input of any word w over S ± 1 , can determine whether w = PL + ( I ) 1?
Algorithmic questions for PL + ( I ) and F , I: The Word Problem Fix a set S = { f 1 , ..., f m } of elements of PL + ( I ). The Word Problem (WP) for the subgroup � S � of PL + ( I ) asks: Q. Is there an algorithm that, upon input of any word w over S ± 1 , can determine whether w = PL + ( I ) 1? Answer 1. If S ⊂ F , the answer is yes.
Algorithmic questions for PL + ( I ) and F , I: The Word Problem Fix a set S = { f 1 , ..., f m } of elements of PL + ( I ). The Word Problem (WP) for the subgroup � S � of PL + ( I ) asks: Q. Is there an algorithm that, upon input of any word w over S ± 1 , can determine whether w = PL + ( I ) 1? Answer 1. If S ⊂ F , the answer is yes. Answer 2. For arbitrary finite S ⊂ PL + ( I ), to solve the Word Problem, multiply the functions. That is, the answer is yes, if... a computer can determine the breakpoints, find slopes to the left and right of breakpoints, multiply slopes, etc.
Computable subgroups of PL + ( I ) , I Let C < PL + ( I ) . h 3 h Split any element g into 2 g = h 1 h − 1 2 h − 1 g 3 into commuting h 1 “one-bump factors” with disjoint support. Spl ( C ) = group generated by the one-bump factors of elements of C . Thm. (Golan, 2016) For any C < PL + ( I ), Spl ( Spl ( C )) = Spl ( C ).
Computable subgroups of PL + ( I ) , II Let C < PL + ( I ) Defn. C is computable if a computer can (a) multiply/invert elements in Spl ( C ), (b) determine breakpoints and endpoints of components of support, and compute the action of Spl ( C ) at those points, (c) compute slopes at support endpoints and the one-bump factors of elements of Spl ( C ), and (d) multiply /invert elements of the group of slopes of one-bump factors at a common endpoint, determine whether this group is discrete, and if so, compute the least slope greater than 1. Notes: • If C < PL + ( I ) is computable and finitely generated, then the word problem for C has a solution. • R. Thompson’s group F is computable.
Algorithmic questions, II: The H -Subgroup Membership Problem Fix a finitely generated C = � S � ⊂ PL + ( I ) and a subgroup H < C . The H -Subgroup Membership Problem ( H -SMP) asks: Q. Is there an algorithm that, upon input of any word w over S ± 1 , can determine whether w ∈ H ? Rmk. The Word Problem is the � 1 � -SMP in � S � .
Algorithmic q’s, II: The H -Subgroup Membership Problem, continued Thm. (Guba, Sapir, 1997) If Thompson’s group F = � S � and H < F satisfies H = Spl ( H ) and H is finitely generated, then the H -SMP in F has a solution.
Algorithmic q’s, II: The H -Subgroup Membership Problem, continued Thm. (Guba, Sapir, 1997) If Thompson’s group F = � S � and H < F satisfies H = Spl ( H ) and H is finitely generated, then the H -SMP in F has a solution. Thm. (Bleak, Brough, H, 2015) If C = � S � is a f.g. computable subgroup of PL + ( I ) and H < C is generated by a finite set of one-bump functions in aligned position, then the H -SMP in C has a solution.
Algorithmic q’s, II: The H -Subgroup Membership Problem, continued Thm. (Guba, Sapir, 1997) If Thompson’s group F = � S � and H < F satisfies H = Spl ( H ) and H is finitely generated, then the H -SMP in F has a solution. Thm. (Bleak, Brough, H, 2015) If C = � S � is a f.g. computable subgroup of PL + ( I ) and H < C is generated by a finite set of one-bump functions in aligned position, then the H -SMP in C has a solution. Def. Let T = { f 1 , ..., f m } be a finite set of one-bump functions and A i := Support ( f i ); then T is in aligned position if: ∀ i ∃ p i ∈ A i such that ∀ i > j : A j ∩ A i � = ∅ = ⇒ A j ⊂ ( p i , f i ( p i )).
Algorithmic q’s, II: The H -Subgroup Membership Problem, continued Thm. (Guba, Sapir, 1997) If Thompson’s group F = � S � and H < F satisfies H = Spl ( H ) and H is finitely generated, then the H -SMP in F has a solution. Thm. (Bleak, Brough, H, 2015) If C = � S � is a f.g. computable subgroup of PL + ( I ) and H < C is generated by a finite set of one-bump functions in aligned position, then the H -SMP in C has a solution. Rmk. Aligned position implies that H is a solvable group. Open Q: For many subgroups H of Thompson’s group F (and of other f.g. computable subgroups of PL + ( I )), the H -SMP is an open question.
Solvable groups: Definition and examples Defn. A group G is solvable if there is a finite sequence 1 = G 0 ⊳ G 1 ⊳ · · · ⊳ G n = G such that each G i +1 /G i is abelian. ( n = derived length of G .) Example 1. The Heisenberg group 1 a b | a, b, c ∈ Z H = 0 1 c 0 0 1 Z 2 ⋊ Z = � x, z | xz = zx � ⋊ � y | � = � x, z, y | xz = zx, yxy − 1 = xz − 1 , yzy − 1 = z � . =
Solvable groups: Definition and examples Defn. A group G is solvable if there is a finite sequence 1 = G 0 ⊳ G 1 ⊳ · · · ⊳ G n = G such that each G i +1 /G i is abelian. ( n = derived length of G .) Example 1. The Heisenberg group 1 a b = 0 1 | a, b, c ∈ Z H c 0 0 1 Z 2 ⋊ Z = � x, z | xz = zx � ⋊ � y | � = � x, z, y | xz = zx, yxy − 1 = xz − 1 , yzy − 1 = z � . = Example 2. The restricted wreath product Z ≀ Z = ( ⊕ n ∈ Z Z ) ⋊ Z = ( ⊕ n ∈ Z � x n � ) ⋊ � t � � x n ( n ∈ Z ) , t | x n x m = x m x n , tx n t − 1 = x n +1 ( n, m ∈ Z ) � . = ( Z ≀ Z is fin. gen. but not fin. pres.) H ≮ PL + ( I ) , but Z ≀ Z < PL + ( I ). Note:
Algorithmic q’s, III: The Uniform Subgroup Membership Problem Fix a finitely generated C = � S � ⊂ PL + ( I ). The uniform subgroup membership problem (USMP) for the group C asks: Q. Is there an algorithm that, upon input of any finite set { f 1 , ..., f n } ∪ { w } of words over S ± 1 , can determine whether w ∈ � f 1 , ..., f n � ? Open Q: The USMP is an open question for F (and many other f.g. computable subgroups of PL + ( I )). USMP solution plan: Split up solvable/nonsolvable cases - Step 1 of algorithm: Determine whether � f 1 , ..., f n � is solvable.
Algorithmic questions, IV: The Solvability Recognition Problem Fix a finitely generated C = � S � ⊂ PL + ( I ). The Solvability Recognition Problem (SRP) for the group C asks: Q. Is there an algorithm that, upon input of any finite set { f 1 , ..., f n } of words over S ± 1 , can determine whether the sub- group � f 1 , ..., f n � is solvable? Thm. (Bleak, Brough, H., 2015) If C is a finitely generated computable subgroup of PL + ( I ), then there is an SRP algorithm for C .
Overview • The groups PL + ( I ) and F • Algorithmic questions for PL + ( I ) and F • Solvability of finitely generated subgroups of PL + ( I ) • More on algorithms
Subgroups of PL + ( I ) and F : Background (Brin, Squier, 1985) PL + ( I ) does not contain a non- Thm. abelian free group. Thm. (Guba, Sapir, 1997) Any nonabelian subgroup of F con- tains a copy of Z ≀ Z .
Subgroups of PL + ( I ) and F : Background (Brin, Squier, 1985) PL + ( I ) does not contain a non- Thm. abelian free group. Thm. (Guba, Sapir, 1997) Any nonabelian subgroup of F con- tains a copy of Z ≀ Z . Thm. (Bleak, 2008) G < PL + ( I ) is solvable iff Spl ( G ) is solv- able. 1 Thm. (Bleak, 2008) Let G < PL + ( I ). If Spl ( G ) contains an infinite sequence of . . . . one-bump functions g i satisfying g Support ( g i ) � Support ( g i +1 ) for all i , 1 g then G is not solvable. i 0 1 { g i } is an infinite tower . 0
Recommend
More recommend
