Superrigidity and the classification problem for the torsion-free abelian groups of

finite rank

Simon Thomas Definition 1 G ⩽ ℚ^{n} has rank n iff G contains n linearly independent elements.

Some History:

1937 Baer gave a satisfactory classification of the rank 1 groups.

1938 Kurosh and Malcev independently gave an unsatisfactory classification of the higher rank groups.

Suppose that G is a torsion-free abelian group and that 0 ≠ x ∈ G. For each prime p ∈ ℙ, the p-height of x is defined to be the supremum

h_{x}(p) ∈ ℕ ∪ {∞}

of the natural numbers n such that

(∃y ∈ G)p^{n}y = x

and the characteristic χ(x) of x is defined to be the function

⟨h_{x}(p) | p ∈ ℙ⟩ ∈ (ℕ ∪ {∞})^{ℙ}.

Two functions χ_{1}, χ_{2} ∈ (ℕ ∪ {∞})^{ℙ} are said to be similar or to belong to the same type, written χ_{1} ≡ χ_{2}, iff

(a) χ_{1}(p) = χ_{2}(p) for almost all primes p; and

(b) if χ_{1}(p) ≠ χ_{2}(p), then both χ_{1}(p) and χ_{2}(p) are finite.

The type τ(x) of x is the ≡-class containing χ(x).

If 0 ≠ G ⩽ ℚ, then τ(G) = τ(x), where x is an arbitrary nonzero element of G.

Theorem 2 (Baer 1937) If 0 ≠ G, H ⩽ ℚ, then G ≅ H iff τ(G) = τ(H).

Fix n ≥ 2. Kurosh and Malcev independently found unsatisfactory complete invariants of the form

⟨M_{p} | p ∈ ℙ⟩ M_{p} ∈ GL_{n}(ℚ_{p})

Why unsatisfactory? Because the problem of deciding whether two sequences are equivalent is as difficult as deciding whether the corresponding groups are isomorphic!

Definition 3 R(ℚ^{n}) is the set of all subgroups G ⩽ ℚ^{n} of rank n.

The classification problem for R(ℚ^{n}) can be reduced to that for R(ℚ^{n+1} ) by the explicit map

R(ℚ^{n}) → R(ℚ^{n+1} )
A ↦ A ⊕ ℚ

Question 4 Does there exist an explicit map f : R(ℚ^{n+1} ) → R(ℚ^{n}) such that

A ≅ B iff f(A) ≅ f(B)?

Question 5 Which functions f : ℝ → ℝ are explicit?

An analogue of Church’s Thesis:

EXPLICIT ⊆ BOREL

BOREL = MEASURABLE modulo null X is a standard Borel space; i.e. a complete separable metric space equipped with its σ-algebra of Borel subsets.

e.g. ℝ, [0, 1], ℚ_{p}, ...

Some less obvious examples:

• P(ℚ^{n}) = 2^{ℚ^{n}}

• The space R(ℚ^{n}) of rank n groups

• The space of finitely generated groups

• The space of countable groups

Theorem 6 There exists a unique uncountable standard Borel space up to isomorphism.

Definition 7 Let E, F be equivalence relations on X, Y.

(a) E ≤_{B} F iff there exists a Borel function

f : X → Y such that

xEy ⟺ f(x)F f(y). f is called a Borel reduction from E to F.

(b) E ∼_{B} F iff both E ≤_{B} F and F ≤_{B} E.

(c) E <_{B} F iff both E ≤_{B} F and E ≁_{B} F. Let ≅_{n} denote the isomorphism relation on R(ℚ^{n}). Then:

(≅_{1}) ≤_{B} \underbrace{(≅_{2}) ≤_{B} (≅_{3}) ≤_{B} (≅_{4}) ≤_{B} ⋯}_{Hard ?}

Using the theory of Borel equivalence relations

• ≅_{1} is the <_{B}-least nontrivial classification problem.

• Let E_{∞} denote the isomorphism relation on the space of finitely generated groups. Then (≅_{n}) ≤_{B} E_{∞} for all n ≥ 1.

Conjecture 8 (Hjorth-Kechris) (≅_{2}) ∼_{B} (E_{∞})

Note that if A, B ∈ R(ℚ^{n}), then

A ≅ B iff ∃g ∈ GL_{n}(ℚ) g(A) = B.

We want to show that there doesn’t exist a Borel map

f : R(ℚ^{n+1} ) → R(ℚ^{n})

A, B lie in the same GL_{n+1} (ℚ)-orbit iff f(A), f(B) lie in the same GL_{n}(ℚ)-orbit.

Lattices in Lie Groups:

Let G be a connected simple Lie group with ℝ-rank (G) ≥ 2.

e.g. SL_{m}(ℝ) for m ≥ 3.

Let ν be a fixed Haar measure on G. Then Δ ⩽ G is a lattice iff

• Δ is a discrete subgroup

• ν(G/Δ) < ∞.

e.g. SL_{m}(ℤ) is a lattice in SL_{m}(ℝ).

A suggestive non-example: Let G = ℝ ⊕ ℝ and Δ = ℤ ⊕ ℤ. Then G/Δ is the torus.

The Measure Context (Zimmer): Let G be a connected centreless simple Lie group with ℝ-rank (G) ≥ 2 and let Δ be a lattice in G. Suppose that

• X is a standard Borel Δ-space.

• Δ acts freely on X.

• μ is a Δ-invariant ergodic probability measure on X.

• E_{Δ}^{X} is the orbit equivalence relation on X.

Question 9 To what extent does the data

(X, E_{Δ}^{X}, μ)

“remember” the group Δ and its action on X?

Definition 10 (X_{0}, Δ_{0}, μ_{0}) and (X_{1}, Δ_{1}, μ_{1}) are orbit equivalent iff there exist

• Δ_{i}-invariant Borel subsets Y_{i} ⊆ X_{i} with μ_{i}(Y_{i}) = 1

• a measure preserving Borel bijection f : Y_{0} → Y_{1}

such that for all a, b ∈ Y_{0},

Δ_{0} · a = Δ_{0} · b ⟺ Δ_{1} · f(a) = Δ_{1} · f(b).

Superrigidity for optimists: Does orbit equivalence imply isomorphism?

Unfortunately, not ...

Theorem 11 (Connes-Feldman-Weiss) If H_{0}, H_{1} are amenable, then (X_{0}, H_{0}, μ_{0}) and (X_{1}, H_{1}, μ_{1}) are orbit equivalent.

However ...

Theorem 12 (Zimmer Superrigidity) With the above hypotheses, if (X_{0}, Δ_{0}, μ_{0}) and (X_{1}, Δ_{1}, μ_{1}) are orbit equivalent, then G_{0} ≅ G_{1}.

A reminder ...

We want to show that there doesn’t exist a Borel map

f : R(ℚ^{n+1} ) → R(ℚ^{n})

A, B lie in the same GL_{n+1} (ℚ)-orbit iff f(A), f(B) lie in the same GL_{n}(ℚ)-orbit.

Unfortunately ...

• GL_{n}(ℚ) is not a lattice.

• There does not exist a GL_{n}(ℚ)-invariant probability measure on R(ℚ^{n}).

• GL_{n}(ℚ) does not act freely on R(ℚ^{n}).

But we can recover ...

Suppose that f : R(ℚ^{n+1} ) → R(ℚ^{n}) is such a Borel map.

• Consider the action of SL_{n+1} (ℤ) on R(ℚ^{n+1} ).

• There exists an ergodic SL_{n+1} (ℤ)-invariant probability measure μ on R(ℚ^{n+1} ). In fact, there exists a class of p-local groups R^{∗}(ℚ^{n+1} ) ⊆ R(ℚ^{n+1} ) such that

(R^{∗}(ℚ^{n+1} ), GL_{n+1} (ℚ))

is essentially the same as

(P G(n, ℚ_{p}), GL_{n+1} (ℚ)).

• Slightly oversimplifying ... reduce to the case when the stabiliser of each f(A) is a fixed group L. Then we obtain a free action of N(L)/L.

Borel analogue of Zimmer superrigidity: Suppose that there is a Borel reduction

f : R(ℚ^{n+1} ) → R(ℚ^{n}).

Then P SL_{n+1} (ℝ) is involved in P SL_{n}(ℝ); i.e. there exist Lie subgroups

N ⊴ H ⩽ P SL_{n}(ℝ)

such that

H/N ≅ P SL_{n+1} (ℝ),

which is a contradiction!

Theorem 13 (S.T. 2000) The complexity of the classification problem for the torsion-free abelian groups of rank n increases strictly with n.

Actually we also need ...

Theorem 14 (Hjorth 1998) Rank 1 groups are strictly easier than rank 2 groups.

The essential point ...

GL_{1}(ℚ) = ℚ^{∗} is amenable.

GL_{2}(ℚ) is nonamenable.

Question 15 Is the classification problem for R(ℚ^{2}) “genuinely difficult”?

Question 16 Does there exist a Borel equivalence relation E such that

(≅_{1}) <_{B} E <_{B} (≅_{2})?

Definition 17 An abelian group A is said to be p-local iff A is q-divisible for every prime q ≠ p.

Definition 18 R^{(p)}(ℚ^{n}) is the standard Borel space of all p-local subgroups G ⩽ ℚ^{n} of rank n.

Definition 19 The isomorphism relation on R^{(p)}(ℚ^{n}) is denoted by ≅_{n}^{(p)} .

Conjecture 20 (S.T. 2000) If p ≠ q are distinct primes, then ≅_{2}^{(p)} and ≅_{2}^{(p)} are incomparable with respect to Borel reducibility. Hence

(≅_{1}) <_{B} (≅_{2}^{(p)}) <_{B} (≅_{2}).

Theorem 21 For each prime p, let E^{p} be the orbit equivalence relation arising from the natural action of GL_{2}(ℚ) on the projective line ℚ_{p} ∪ {∞}.

(\begin{matrix}a & b \\ c & d \end{matrix}) · x = \frac{ax + b}{cx + d}

Then ≅_{2}^{(p)}∼_{B} E^{p}.

Theorem 22 (S.T. 2002) Fix some n ≥ 3. If p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank n are incomparable with respect to Borel reducibility.

Makes use of

• Zimmer superrigidity for SL_{n}(ℝ).

• The Ratner measure classification theorem for real Lie groups.

• The Kazhdan property for SL_{n}(ℤ).

Theorem 23 (Hjorth-S.T. 2004) If p ≠ q are distinct primes, then ≅_{2}^{(p)} and ≅_{2}^{(p)} are incomparable with respect to Borel reducibility. Hence

(≅_{1}) <_{B} (≅_{2}^{(p)}) <_{B} (≅_{2}).

Makes use of

• Zimmer superrigidity for

SL_{2}(ℝ) × SL_{2}(ℚ_{p}).

• The Margulis-Tomanov measure classification theorem for products of real and p-adic Lie groups.

• Property (τ) for SL_{2}(ℤ[1/p]).