The Homology of Richard Thompson's Group F Abstract 〈X0,X1,X2,... ; Xxi

The homology of Richard Thompson's group F

Ken Brown Cornell University

Abstract

Let F be Thompson’s group, with presentation

xi h x0, x1, x2,... ; xn = xn+1 for i < n i .

Geoghegan and I showed in the early 1980s that Hn(F ) is free abelian of rank 2 for all n ≥ 1. It turns out that the homology admits a natural ring structure, which I calculated a few years later but never published. With the aid of this ring structure one can also calculate the cohomology ring. In this talk, which is dedicated to Ross Geoghegan in honor of his 60th birthday, I will explain where the ring structure comes from and describe the method of calculation.

1 xi Review of F = h x0,x1,x2,... ; xn = xn+1 (i < n) i

1. (Thompson) F is the group of dyadic PL-homeomorphisms of I.

2. (Freyd–Heller, Dydak) F is the universal example of a group with an endomorphism that is idempotent up to conjugacy:

2 x0 φ(xn)= xn+1 , φ = φ

Homeomorphism interpretation: φ(f)= “f concentrated on [1/2, 1]”.

3. (Galvin–Thompson) F is isomorphic to the group of order-preserving automorphisms of a free Cantor algebra:

µ: X × X → X (bijection)

4. F is ﬁnitely generated (x0, x1).

x0x0 x0x1 x0x0x0 x0x0x1 5. F is ﬁnitely presented (x1 = x1 , x1 = x1 ). 6. (Brown–Geoghegan) And so on.

2 F is the group of dyadic PL-homeomorphisms of I

All slopes are integral powers of 2, all breakpoints have dyadic rational coordinates.

3 A product on F

There is a homomorphism

F × F → F (f, g) 7→ f ∗ g , which is associative up to conjugacy:

f ∗ (g ∗ h)=((f ∗ g) ∗ h)x0 .

By deﬁnition, f ∗ g = f on [0, 1/2] and g on [1/2, 1] .

There’s no identity: 1 ∗ f = φ(f) .

4 x1

x0 ∗ x1

5 H∗(F ) as a ring

The ∗-product F × F → F makes H∗(F ) an associative ring. No identity; left multiplication by the canonical generator e ∈ H0(F )= Z is a rank 1 idempotent in each positive dimension, equal to φ∗. Right multiplication by e is a different rank 1 idempotent.

Theorem 1. H∗(F ) is an associative algebra (without identity) generated by e (degree 0), α,β (degree 1), subject to the relations

e2 = e eα = βe =0 αe = α , eβ = β

Consequence: α2 = β2 =0, alternating products αβα ··· and βαβ ··· give basis in positive dimensions.

6 ∗ H (F ) as a ring ∗ H (F ) is dual to H∗(F ), which is a coalgebra in the usual way; the diagonal F → F × F induces an algebra homomorphism

∆: H∗(F ) → H∗(F ) ⊗ H∗(F ) .

∗ Theorem 2. H (F ) =∼ (a, b) ⊗ Γ(u), where deg a = deg b =1, deg u =2. V

Here Γ(u) is a divided polynomial ring: the subring of Q[u] spanned additively by the elements u(i) := ui/i! (i ≥ 0). Note ui+j i + j u(i)u(j) = = u(i+j) i!j! i

Can also state result in form: ∗ ∼ ∗ 0 ∗ ∗ 0 ∼ H (F ) = H (F ) ⊗ H (Fab) , H (F ) = Γ(u)

7 Sketch of proof of Theorem 1

By variant of Brown–Geoghegan, get a cubical K(F, 1) such that set of cubes is free semigroup (without 1) on a 0-cube v and a 1-cube c with vertices v and v2.

Resulting chain complex C∗ = C∗(F ) is free associative ring (i.e.,

noncommutative polynomial ring), without 1, on v ∈ C0 and c ∈ C1, with ∂c = v2 − v.

Equivalently: C∗ is universal example of a differential graded Z-algebra with an element v of degree 0 such that v2 ' v.

Now have 1-cycles z := vc − cv and zv. Can check, either by B–G arguments or direct algebraic argument, that v,z,zv generate homology and give the stated calculation.

See Stein [4] for details and a generalization.

Note: e = v, α = zv, β = z − α.

8 Construction of cubical K(F, 1) 1. Let F be category of intervals [0, n] (n =1, 2,... ) and dyadic maps. Then BF ' BF = original K(F, 1) constructed by Eilenberg and MacLane. Have strictly associative product F×F→F gotten by gluing intervals together. Note objects are the powers of v := [0, 1]. 2. Have homotopy equivalent subcategory T ,→ F consisting of dyadic subdivision maps [0, n] → [0, k] (n ≥ k). 3. Desired cubical complex is homotopy equivalent subcomplex of BT built using simple subdivisions. Example: Given 1 ≤ i < j ≤ n, there’s a square corresponding to halving the ith and jth subintervals of [0, n]:

γ [0, n + 2] i [0, n + 1]

γj+1 γj

[0, n + 1] [0, n] γi

9 References

[1] K. S. Brown and R. Geoghegan, FP∞ groups and HNN extensions, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 227–229. MR 707963

[2] , An inﬁnite-dimensional torsion-free FP∞ group, Invent. Math. 77 (1984), no. 2, 367–381. MR 752825

[3] , Cohomology with free coefﬁcients of the fundamental group of a graph of groups, Comment. Math. Helv. 60 (1985), no. 1, 31–45. MR 787660

[4] M. Stein, Groups of piecewise linear homeomorphisms, Trans. Amer. Math. Soc. 332 (1992), no. 2, 477–514. MR 1094555