SLIDE 1
- 1. Thompson’s definition
F < T < V F is the group of associative laws, T allows cyclic - - PowerPoint PPT Presentation
F < T < V F is the group of associative laws, T allows cyclic - - PowerPoint PPT Presentation
Introduction to Thompsons group Ken Brown Cornell University Abstract Forty years ago Richard Thompson introduced a fascinating discrete group F , which has become a test case for many questions in geometric group theory. I will describe F
SLIDE 2
SLIDE 3
Composition a(b(c(de)))
x0 x2 x1x0
(ab)(c(de))
x1
a(b((cd)e))
x0
(ab)((cd)e)
A relation:
x1x0 = x0x2
- r
xx0
1 = x2
More generally,
xnxi = xixn+1
- r
xxi
n = xn+1
(i < n)
Fact: x0, x1, x2, . . . generate F , and these are defining relations.
3
SLIDE 4
- 2. Combinatorial group theory
F = x0, x1, x2, . . . ; xxi
n = xn+1 for i < n
xnxi → xixn+1
(smaller subscripts first)
x−1
i xn → xn+1x−1 i
(positive before negative)
x−1
n xi → xix−1 n+1
(positive before negative)
x−1
i x−1 n
→ x−1
n+1x−1 i
(smaller subscripts last) Normal forms:
f = xi1xi2 · · · xikx−1
jl · · · x−1 j2 x−1 j1
(i1 ≤ · · · ≤ ik , j1 ≤ · · · ≤ jl)
Unique if reduced: If xi and x−1
i
both occur, then so does xi+1 or x−1
i+1.
x0x1x1 x3x−1
5 x−1 4
x−1
1 x−1
= x0x1 x2x−1
4 x−1 3
x−1
4
SLIDE 5
- 3. Group of fractions
F is the group of right fractions of its positive semigroup P : f ∈ F = ⇒ f = pq−1 (p, q ∈ P) P has a concrete interpretation as the semigroup of binary forests (Belk, Brin).
5
SLIDE 6
· · · · · · x0 x1 · · · x2
6
SLIDE 7
· · · x0 x1 · · · · · · x0x1
7
SLIDE 8
Relations · · · x1x0 = x0x2 x2x0 = x0x3 · · ·
8
SLIDE 9
- 4. Dyadic PL-homeomorphisms of I (or R+ or R)
F ∼ = PL2(I) [∼ = PL2(R+) ∼ = PL2(R)]. All slopes are integral powers of 2, all
breakpoints have dyadic rational coordinates, integer translation near ±∞ if use
R+ or R.
9
SLIDE 10
PL2(I) ∼ = PL2(R+) ∼ = PL2(R)
10
SLIDE 11
- 5. Tree and forest diagrams
Binary trees encode binary subdivisions or parenthesized expressions. If use R+, get forest diagrams (but we knew this already). If use R, get doubly-infinite forest diagrams.
11
SLIDE 12
- 6. Universal conjugacy idempotent
(Freyd–Heller, Dydak) F is the universal example of a group with an endomorphism that is idempotent up to conjugacy:
φ(xn) = xn+1 , φ2 = φx0
Homeomorphism interpretation: φ(f) = “f concentrated on [1/2, 1]”. Universality: Given any φ: G → G with φ2 conjugate to φ, need x0 so that (1)
φ2 = φx0 ,
then need x1 = φ(x0), x2 = φ(x1),. . . . Equation (1) forces
xn+1 = xx0
n
(n > 0) ,
apply φ to get remaining relations.
12
SLIDE 13
- 7. Algebra automorphisms
(Galvin–Thompson) F is isomorphic to the group of order-preserving automorphisms of a free Cantor algebra:
µ: X × X → X
(bijection) Everything splits uniquely as a product.
a = a0a1 = a0(a10a11) a = a0a1 = (a00a01)a1
Every tree diagram (or associative law) gives an automorphism.
a a0 a1 a10 a11 a00 a01 a1 a a0 x0
13
SLIDE 14
Why is F interesting?
- Comes up in many ways.
- Has interesting properties.
- Almost every question is a challenge.
14
SLIDE 15
Known properties of F
- 1. Good finiteness properties: Two generators x0, x1. Two relations
xx0x0
1
= xx0x1
1
and xx0x0x0
1
= xx0x0x1
1
. And so on (Brown–Geoghegan).
- 2. F is “almost simple”.
1 → F ′ → F → Z × Z → 0
- 3. Although highly nonabelian, F admits a product F × F → F , associative
up to conjugacy. [No identity; 1 ∗ 1 = 1, but 1 ∗ f = φ(f) in general.]
- 4. F has no free subgroups.
- 5. F is not elementary amenable.
- 6. Isoperimetric constant with respect to x0, x1 is ≤ 1/2 (Belk–Brown).
- 7. The Poisson boundary for (some) symmetric random walks is nontrivial
(Kaimanovich).
- 8. Homology is known (B–G): Hn(F) ∼
= Z ⊕ Z for all n ≥ 1.
15
SLIDE 16
- 9. Homology and cohomology are known as rings (B): H∗(F) is an associative
algebra (without identity) generated by e (degree 0), α, β (degree 1), subject to relations
e2 = e eα = βe = 0 αe = α , eβ = β
Consequence: α2 = β2 = 0, alternating products αβα · · · and βαβ · · · give basis in positive dimensions. H∗(F) ∼
= (a, b) ⊗ Γ(u).
- 10. F is orderable.
- 11. Easy algorithm for computing length function (Fordham, Belk–Brown).
- 12. Growth series explicitly known for P (Burillo, B–B):
p(x) = 1 − x2 1 − 2x − x2 + x3
16
SLIDE 17
Open problems
- 1. Is F amenable?
- 2. Is the isoperimetric constant 1/2? Is it ≥ 1/2 for all generating sets?
- 3. Describe the Poisson boundary or other invariants of random walk.
- 4. Is F automatic?
- 5. What is the exponential growth rate of F ?
- 6. Is the growth series rational?
- 7. Is it true that every subgroup of F is either elementary amenable or contains