Algorithms for groups of homeomorphisms Susan Hermiller Joint work - - PowerPoint PPT Presentation

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].

• 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 {2n | n ∈ Z} and breakpoints in Z[1

2].

F = x0, x1 | [x0x−1

1 , x−1 0 x1x0], [x0x−1 1 , x−2 0 x1x2 0]

is finitely presented. x0 =

      

2x ifx ∈ [0, 1

4]

x + 1

4

if x ∈ [1

4, 1 2] 1 2x + 1 2

if x ∈ [1

2, 1]

x1 =

            

x if x ∈ [0, 1

2]

2x − 1

2

ifx ∈ [1

2, 5 8]

x + 1

8

if x ∈ [5

8, 3 4] 1 2x + 1 2

if x ∈ [3

4, 1]

• 1/2

The groups PL+(I) and F

• Rmk. There are many ways to represent an element g
• f 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 = {f1, ..., fm} 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 = {f1, ..., fm} 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 = {f1, ..., fm} 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). Split any element g into into commuting “one-bump factors” with disjoint support.

g h h h 1 2 3

g = h1h−1

2 h−1 3

Spl(C) = group generated by the one-bump factors of elements

• f 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

• ne-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

• ne-bump functions in aligned position, then the H-SMP in C

has a solution.

• Def. Let T = {f1, ..., fm} be a finite set of one-bump functions

and Ai := Support(fi); then T is in aligned position if: ∀ i ∃ pi ∈ Ai such that ∀ i > j: Aj ∩ Ai = ∅ = ⇒ Aj ⊂ (pi, fi(pi)).

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

• ne-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

• f 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 = G0 ⊳ G1 ⊳ · · · ⊳ Gn = G such that each Gi+1/Gi is abelian. (n = derived length of G.) Example 1. The Heisenberg group H =

       

1 a b 1 c 1

   | a, b, c ∈ Z     

= Z2 ⋊ 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 = G0 ⊳ G1 ⊳ · · · ⊳ Gn = G such that each Gi+1/Gi is abelian. (n = derived length of G.) Example 1. The Heisenberg group H =

       

1 a b 1 c 1

   | a, b, c ∈ Z     

= Z2 ⋊ 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∈ZZ) ⋊ Z = (⊕n∈Zxn) ⋊ t = xn (n ∈ Z), t | xnxm = xmxn, txnt−1 = xn+1 (n, m ∈ Z). (Z ≀ Z is fin. gen. but not fin. pres.) Note: H ≮ PL+(I) , but Z ≀ Z < PL+(I).

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

{f1, ..., fn} ∪ {w} of words over S±1, can determine whether w ∈ f1, ..., fn? 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 f1, ..., fn 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

{f1, ..., fn} of words over S±1, can determine whether the sub- group f1, ..., fn 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 Thm. (Brin, Squier, 1985) PL+(I) does not contain a non- 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 Thm. (Brin, Squier, 1985) PL+(I) does not contain a non- 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.

• Thm. (Bleak, 2008) Let G < PL+(I). If

Spl(G) contains an infinite sequence of

• ne-bump functions gi satisfying

Support(gi) Support(gi+1) for all i, then G is not solvable. {gi} is an infinite tower.

. . . . g g

Solvability of fin. gen. subgroups of PL+(I), I

• Thm. (Bleak, Brough, H., 2015)

If S is a finite subset of PL+(I) and G = S is solvable, then the derived length of G is at most the number N of breakpoints

• f the elements of S.

Solvability of fin. gen. subgroups of PL+(I), I

• Thm. (Bleak, Brough, H., 2015)

If S is a finite subset of PL+(I) and G = S is solvable, then the derived length of G is at most the number N of breakpoints

• f the elements of S.

Note: This (finite) bound on the derived length allows the Solv- ability Recognition Problem algorithm to halt when the group is not solvable.

Solvability of 2-gen. subgroups of PL+(I), I

• Plan. Consider a subgroup of PL+(I) generated by two
• ne-bump functions f, g.

There are five possible configurations: (1) = no overlap

f g

Support(f) ∩ Support(g) = ∅. f and g commute. f, g is solvable (abelian).

Solvability of 2-gen. subgroups of PL+(I), II (2) = “one-sided overlap”:

f g

• Lemma. Support(g−1fg) =

g(Support(f))

• {g−ifgi} is an infinite tower; hence f, g and F are not solvable.

Solvability of 2-gen. subgroups of PL+(I), III (3) = “chain”

g f

{f, g−1fg} is a one-sided overlap (configuration (2)) Hence f, g is not solvable.

Solvability of 2-gen. subgroups of PL+(I), IV (4)

g f

(Support(g−1fg) = g(Support(f))) Cases: (4a) If Support(f) ∩ Support(fg) = ∅: Then f, fg ∈ f, g give a chain (configuration (3)); f, g is not solvable. (4b) If Support(f) ∩ Support(fg) = ∅: f, g = (⊕n∈ZZ) ⋊ Z = Z ≀ Z (with n-th Z = fgn, and top Z = g) is solvable.

g f f f g g

Solvability of 2-gen. subgroups of PL+(I), V (5)

g f

(5a) If there is a one-bump c ∈ f, g with Support(c) = Support(f) = (a, b), and integers k, ℓ such that f′

+(a) = k · c′ +(a), f′ −(b) =

k · c′

−(b), g′ +(a) = ℓ · c′ +(a), g′ −(b) = ℓ · c′ −(b):

One-bump factors of {c, f · c−k} or {c, g · c−ℓ} give new towers. One-bump factors from c, f · c−k, g · c−ℓ are in configurations (1), (2), (3), or (4).

• k

(5b) If there are no such c, k, l: Then f, g is not solvable.

Solvability of fin. gen. subgroups of PL+(I), II Thm 3. (Bleak, Brough, H., 2015) Let G < PL+(I) be a finitely generated group. The following are equivalent: (1) G is solvable (2) Spl(G) does not contain a chain. (3) Spl(G) does not contain a one-sided overlap. (4) Spl(G) does not contain an infinite tower.

Solvability of fin. gen. subgroups of PL+(I), II Thm 3. (Bleak, Brough, H., 2015) Let G < PL+(I) be a finitely generated group. The following are equivalent: (1) G is solvable (2) Spl(G) does not contain a chain. (3) Spl(G) does not contain a one-sided overlap. (4) Spl(G) does not contain an infinite tower. (5) G does not contain a copy of the Brin-Navas group. The Brin-Navas group B = {wi | i ∈ Z}, s | ws

i = wi+1, [w wm

k

i

, wj] = 1 (i < k, j < k, m ∈ Z \ {0}) is finitely generated by {s, w0} and contains a copy of every solv- able subgroup of PL+(I).

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

Solvability Recognition Problem Algorithm Recall:

• Thm. (Bleak, Brough, H., 2015)

If C = S is a finitely generated computable subgroup of PL+(I), then there is an SRP algorithm for C; that is, upon input of any finite set {f1, ..., fn} of words over S±1, the algorithm determines whether the subgroup f1, ..., fn is solvable. Algorithm overview: Process overlaps of components of support (a, b), (c, d) among the one-bump factors of the generators for C. While building ever deeper towers from overlaps of type (5) [a = c and b = d], search for (2) one-sided overlaps [a = c < b < d] and (3) chains [a < c < b < d]. Either reach a tower deeper than (# breakpoints of generators

• f G) [then not solvable], or run out of elements for building

deeper towers [solvable].