The isomorphism and bi-embeddability relations for finitely - - PowerPoint PPT Presentation

The isomorphism and bi-embeddability relations for finitely generated groups

Simon Thomas

Rutgers University

1st June 2016

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The starting point ...

Theorem (Thomas-Velickovic 1998)

The isomorphism relation ∼ = on the space Gfg of finitely generated groups is countable universal.

Theorem (Williams 2012)

The bi-embeddability relation ≈ on the space Gfg of finitely generated groups is countable universal.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The starting point ...

Theorem (Thomas-Velickovic 1998)

The isomorphism relation ∼ = on the space Gfg of finitely generated groups is countable universal.

Theorem (Williams 2012)

The bi-embeddability relation ≈ on the space Gfg of finitely generated groups is countable universal.

Definition

A Borel equivalence relation E is countable if every E-class is countable.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The starting point ...

Theorem (Thomas-Velickovic 1998)

The isomorphism relation ∼ = on the space Gfg of finitely generated groups is countable universal.

Theorem (Williams 2012)

The bi-embeddability relation ≈ on the space Gfg of finitely generated groups is countable universal.

Definition

A countable Borel equivalence relation E is universal if F ≤B E for every countable Borel equivalence relation F.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The basic problem

The Basic Problem

Determine the Borel complexity of the isomorphism and bi-embeddability relations for various restricted classes of finitely generated groups, such as:

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The basic problem

The Basic Problem

Determine the Borel complexity of the isomorphism and bi-embeddability relations for various restricted classes of finitely generated groups, such as: simple groups

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The basic problem

The Basic Problem

Determine the Borel complexity of the isomorphism and bi-embeddability relations for various restricted classes of finitely generated groups, such as: simple groups Kazhdan groups

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a simple group?

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a simple group?

Definition

A group G is simple if its only normal subgroups are 1 and G.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a simple group?

Definition

A group G is simple if its only normal subgroups are 1 and G.

Observation

The class of simple groups is axiomatizable by a sentence of Lω1ω.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a simple group?

Definition

A group G is simple if its only normal subgroups are 1 and G.

Observation

The class of simple groups is axiomatizable by a sentence of Lω1ω.

Proof.

A group G is simple if and only if for every 1 = g ∈ G, the conjugacy class gG = { xgx−1 | x ∈ G } generates G.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a Kazhdan group?

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a Kazhdan group?

Definition

A countable group G is a Kazhdan group if every isometric action

• f G on a real Hilbert space has a global fixed point.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a Kazhdan group?

Definition

A countable group G is a Kazhdan group if every isometric action

• f G on a real Hilbert space has a global fixed point.

Theorem (Shalom 2000)

{ G ∈ Gfg | G is Kazhdan } is an open subset of the space Gfg

• f finitely generated groups.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

What is a Kazhdan group?

Definition

A countable group G is a Kazhdan group if every isometric action

• f G on a real Hilbert space has a global fixed point.

Theorem (Shalom 2000)

{ G ∈ Gfg | G is Kazhdan } is an open subset of the space Gfg

• f finitely generated groups.

Corollary (Shalom 2000)

Every finitely generated Kazhdan group is a homomorphic image

• f a finitely presented Kazhdan group.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

A marked group (G, ¯ s) consists of a f.g. group with a distinguished sequence ¯ s = (s1, · · · , sm) of generators.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

A marked group (G, ¯ s) consists of a f.g. group with a distinguished sequence ¯ s = (s1, · · · , sm) of generators. For each m ≥ 1, let Gm be the set of isomorphism types of marked groups (G, (s1, · · · , sm)) with m distinguished generators.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

A marked group (G, ¯ s) consists of a f.g. group with a distinguished sequence ¯ s = (s1, · · · , sm) of generators. For each m ≥ 1, let Gm be the set of isomorphism types of marked groups (G, (s1, · · · , sm)) with m distinguished generators. Then there exists a canonical embedding Gm ֒ → Gm+1 defined by (G, (s1, · · · , sm)) → (G, (s1, · · · , sm, 1G)).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

A marked group (G, ¯ s) consists of a f.g. group with a distinguished sequence ¯ s = (s1, · · · , sm) of generators. For each m ≥ 1, let Gm be the set of isomorphism types of marked groups (G, (s1, · · · , sm)) with m distinguished generators. Then there exists a canonical embedding Gm ֒ → Gm+1 defined by (G, (s1, · · · , sm)) → (G, (s1, · · · , sm, 1G)). And Gfg = Gm is the space of f.g. groups.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Let (G, ¯ s) ∈ Gm and let dS be the corresponding word metric. For each ℓ ≥ 1, let Bℓ(G, ¯ s) = {g ∈ G | dS(g, 1G) ≤ ℓ}.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Let (G, ¯ s) ∈ Gm and let dS be the corresponding word metric. For each ℓ ≥ 1, let Bℓ(G, ¯ s) = {g ∈ G | dS(g, 1G) ≤ ℓ}. The basic open neighborhoods of (G, ¯ s) in Gm are given by U(G,¯

s),ℓ = { (H,¯

t) ∈ Gm | Bℓ(H,¯ t) ∼ = Bℓ(G, ¯ s) }, ℓ ≥ 1.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Let (G, ¯ s) ∈ Gm and let dS be the corresponding word metric. For each ℓ ≥ 1, let Bℓ(G, ¯ s) = {g ∈ G | dS(g, 1G) ≤ ℓ}. The basic open neighborhoods of (G, ¯ s) in Gm are given by U(G,¯

s),ℓ = { (H,¯

t) ∈ Gm | Bℓ(H,¯ t) ∼ = Bℓ(G, ¯ s) }, ℓ ≥ 1.

Observation

The isomorphism and bi-embeddability relations on the space Gfg

• f finitely generated groups are both countable Borel.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Proposition

For each G ∈ Gfg, there exists a sequence of finitely presented groups Gn ∈ Gfg such that: G0 ։ G1 ։ G2 · · · ։ Gn ։ · · · ։ G. G is a homomorphic image of each Gn. limn→∞ Gn = G.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Proposition

For each G ∈ Gfg, there exists a sequence of finitely presented groups Gn ∈ Gfg such that: G0 ։ G1 ։ G2 · · · ։ Gn ։ · · · ։ G. G is a homomorphic image of each Gn. limn→∞ Gn = G.

Proof.

Let G = (G, ¯ s) ∈ Gfg.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Proposition

For each G ∈ Gfg, there exists a sequence of finitely presented groups Gn ∈ Gfg such that: G0 ։ G1 ։ G2 · · · ։ Gn ։ · · · ։ G. G is a homomorphic image of each Gn. limn→∞ Gn = G.

Proof.

Let G = (G, ¯ s) ∈ Gfg. Let Rn be the identities w(¯ s) = 1 which are visible in Bn(G, ¯ s).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Proposition

For each G ∈ Gfg, there exists a sequence of finitely presented groups Gn ∈ Gfg such that: G0 ։ G1 ։ G2 · · · ։ Gn ։ · · · ։ G. G is a homomorphic image of each Gn. limn→∞ Gn = G.

Proof.

Let G = (G, ¯ s) ∈ Gfg. Let Rn be the identities w(¯ s) = 1 which are visible in Bn(G, ¯ s). Then Gn = ¯ s | Rn satisfies our requirements.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Corollary

If K is an open subset of Gfg, then every G ∈ K is a homomorphic image of a finitely presented group H ∈ K.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Polish space of finitely generated groups

Corollary

If K is an open subset of Gfg, then every G ∈ K is a homomorphic image of a finitely presented group H ∈ K.

Proof.

Let ( Gn ) be a sequence of finitely presented groups such that: G is a homomorphic image of Gn. limn→∞ Gn = G. Then Gn ∈ K for all but finitely many n ∈ N.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Main Theorems

Theorem (Thomas-Williams 2013)

The isomorphism and bi-embeddability relations on the space of Kazhdan groups are both weakly universal.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Main Theorems

Theorem (Thomas-Williams 2013)

The isomorphism and bi-embeddability relations on the space of Kazhdan groups are both weakly universal.

Theorem (Thomas 2013)

The isomorphism and bi-embeddability relations on the space of finitely generated simple groups are both nonsmooth.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The Main Theorems

Theorem (Thomas-Williams 2013)

The isomorphism and bi-embeddability relations on the space of Kazhdan groups are both weakly universal.

Theorem (Thomas 2013)

The isomorphism and bi-embeddability relations on the space of finitely generated simple groups are both nonsmooth.

Definition

The Borel equivalence relation E is smooth if E is Borel reducible to the identity relation ∆X on some (equivalently every) uncountable Polish space X.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Weakly universal Borel equivalence relations

Definition

A countable Borel equivalence relation E is weakly universal if for every countable Borel equivalence relation F, there exists a countable-to-one Borel homomorphism from F to E.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Weakly universal Borel equivalence relations

Definition

A countable Borel equivalence relation E is weakly universal if for every countable Borel equivalence relation F, there exists a countable-to-one Borel homomorphism from F to E.

Theorem (Kechris-Miller 2008)

A countable Borel equivalence relation E is weakly universal iff there exists a universal countable Borel equivalence relation F ⊆ E.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Weakly universal Borel equivalence relations

Definition

A countable Borel equivalence relation E is weakly universal if for every countable Borel equivalence relation F, there exists a countable-to-one Borel homomorphism from F to E.

Example

The Turing equivalence relation ≡ T on 2N is weakly universal.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Weakly universal Borel equivalence relations

Question

Does there exist a weakly universal Borel equivalence relation which is not countable universal?

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Weakly universal Borel equivalence relations

Question

Does there exist a weakly universal Borel equivalence relation which is not countable universal?

Theorem (Dougherty & Kechris 1999)

If Martin’s Conjecture on degree invariant Borel maps holds, then the Turing equivalence relation ≡ T is weakly universal but not countable universal.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The main theorems

Theorem (Thomas-Williams 2013)

The isomorphism and bi-embeddability relations on the space of Kazhdan groups are both weakly universal.

Theorem (Thomas 2013)

The isomorphism and bi-embeddability relations on the space of finitely generated simple groups are both nonsmooth.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The main theorems

Theorem (Thomas-Williams 2013)

The isomorphism and bi-embeddability relations on the space of Kazhdan groups are both weakly universal.

Theorem (Thomas 2013)

The isomorphism and bi-embeddability relations on the space of finitely generated simple groups are both nonsmooth.

Initial Target

The bi-embeddability relation ≈ on the space Gfg of finitely generated groups is weakly universal.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The word problem for finitely generated groups

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The word problem for finitely generated groups

For each m ≥ 1, fix a recursive enumeration { wk(x1, · · · , xm) | k ∈ N }

• f the (not necessarily reduced) words in x1, · · · , xm, x−1

1 , · · · , x−1 m .

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The word problem for finitely generated groups

For each m ≥ 1, fix a recursive enumeration { wk(x1, · · · , xm) | k ∈ N }

• f the (not necessarily reduced) words in x1, · · · , xm, x−1

1 , · · · , x−1 m .

Definition

If (G, ¯ s) ∈ Gm is a finitely generated group, then Rel(G) = { k ∈ N | wk(s1, · · · , sm) = 1 } Nonrel(G) = { k ∈ N | wk(s1, · · · , sm) = 1 }.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The word problem for finitely generated groups

For each m ≥ 1, fix a recursive enumeration { wk(x1, · · · , xm) | k ∈ N }

• f the (not necessarily reduced) words in x1, · · · , xm, x−1

1 , · · · , x−1 m .

Definition

If (G, ¯ s) ∈ Gm is a finitely generated group, then Rel(G) = { k ∈ N | wk(s1, · · · , sm) = 1 } Nonrel(G) = { k ∈ N | wk(s1, · · · , sm) = 1 }.

Question

But what if we choose a different generating set G = t1, · · · , tn ?

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The most obvious Turing reductions

Definition

If A, B ∈ 2N, then A is one-one reducible to B, written A ≤1 B, if there exists an injective recursive function f : N → N such that for all n ∈ N, n ∈ A ⇐ ⇒ f(n) ∈ B.

Definition

If A ≤1 B and B ≤1 A, then we write A ≡1 B and say that A, B are recursively isomorphic.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The most obvious Turing reductions

Definition

If A, B ∈ 2N, then A is one-one reducible to B, written A ≤1 B, if there exists an injective recursive function f : N → N such that for all n ∈ N, n ∈ A ⇐ ⇒ f(n) ∈ B.

Definition

If A ≤1 B and B ≤1 A, then we write A ≡1 B and say that A, B are recursively isomorphic.

Example

If (G, ¯ s), (H,¯ t) ∈ Gfg and G ֒ → H, then Rel(G, ¯ s) ≤1 Rel(H,¯ t) and Nonrel(G, ¯ s) ≤1 Nonrel(H,¯ t).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal finitely generated groups

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal finitely generated groups

Definition

If G is any group, then the corresponding skeleton is defined to be Sk(G) = { H ∈ Gfg | H ֒ → G }.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal finitely generated groups

Definition

If G is any group, then the corresponding skeleton is defined to be Sk(G) = { H ∈ Gfg | H ֒ → G }.

Definition

If A ∈ 2N, then the finitely generated group G is relatively universal

• f degree A if Sk(G) = { H ∈ Gfg | Rel(H) ≤T A }.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal finitely generated groups

Definition

If G is any group, then the corresponding skeleton is defined to be Sk(G) = { H ∈ Gfg | H ֒ → G }.

Definition

If A ∈ 2N, then the finitely generated group G is relatively universal

• f degree A if Sk(G) = { H ∈ Gfg | Rel(H) ≤T A }.

Observation

If A, B ∈ 2N and GA, GB are relatively universal of degrees A, B, then GA ≈ GB ⇐ ⇒ A ≡T B.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal finitely generated groups

Definition

If G is any group, then the corresponding skeleton is defined to be Sk(G) = { H ∈ Gfg | H ֒ → G }.

Definition

If A ∈ 2N, then the finitely generated group G is relatively universal

• f degree A if Sk(G) = { H ∈ Gfg | Rel(H) ≤T A }.

Theorem

For each f.g. group G, there exists a f.g. group H such that: Rel(H) ≤ T Rel(G); and H does not embed into G.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an .

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an . Let { uk(x1, · · · , xn), vk(x1, · · · , xn) | k ∈ N } be a recursive enumeration of the ordered pairs of words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an . Let { uk(x1, · · · , xn), vk(x1, · · · , xn) | k ∈ N } be a recursive enumeration of the ordered pairs of words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

For each k ∈ N, let rk(x, y) be the word ( xk+1yk+1 )7.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an . Let { uk(x1, · · · , xn), vk(x1, · · · , xn) | k ∈ N } be a recursive enumeration of the ordered pairs of words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

For each k ∈ N, let rk(x, y) be the word ( xk+1yk+1 )7. Let H have presentation b, c | R , where rk(b, c) ∈ R ⇐ ⇒ rk(uk(a1, · · · , an), vk(a1, · · · , an)) = 1

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an . Let { uk(x1, · · · , xn), vk(x1, · · · , xn) | k ∈ N } be a recursive enumeration of the ordered pairs of words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

For each k ∈ N, let rk(x, y) be the word ( xk+1yk+1 )7. Let H have presentation b, c | R , where rk(b, c) ∈ R ⇐ ⇒ rk(uk(a1, · · · , an), vk(a1, · · · , an)) = 1 “Clearly” Rel(H) ≤ T Rel(G).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an . Let { uk(x1, · · · , xn), vk(x1, · · · , xn) | k ∈ N } be a recursive enumeration of the ordered pairs of words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

For each k ∈ N, let rk(x, y) be the word ( xk+1yk+1 )7. Let H have presentation b, c | R , where rk(b, c) ∈ R ⇐ ⇒ rk(uk(a1, · · · , an), vk(a1, · · · , an)) = 1 “Clearly” Rel(H) ≤ T Rel(G). If ϕ : H ֒ → G, then there exists k ∈ N such that ϕ(b) = uk(a1, · · · , an) and ϕ(c) = vk(a1, · · · , an).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an . Let { uk(x1, · · · , xn), vk(x1, · · · , xn) | k ∈ N } be a recursive enumeration of the ordered pairs of words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

For each k ∈ N, let rk(x, y) be the word ( xk+1yk+1 )7. Let H have presentation b, c | R , where rk(b, c) ∈ R ⇐ ⇒ rk(uk(a1, · · · , an), vk(a1, · · · , an)) = 1 “Clearly” Rel(H) ≤ T Rel(G). If ϕ : H ֒ → G, then there exists k ∈ N such that ϕ(b) = uk(a1, · · · , an) and ϕ(c) = vk(a1, · · · , an). Clearly rk(b, c) / ∈ R and so rk(uk(a1, · · · , an), vk(a1, · · · , an)) = 1.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let G = a1, · · · , an . Let { uk(x1, · · · , xn), vk(x1, · · · , xn) | k ∈ N } be a recursive enumeration of the ordered pairs of words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

For each k ∈ N, let rk(x, y) be the word ( xk+1yk+1 )7. Let H have presentation b, c | R , where rk(b, c) ∈ R ⇐ ⇒ rk(uk(a1, · · · , an), vk(a1, · · · , an)) = 1 “Clearly” Rel(H) ≤ T Rel(G). If ϕ : H ֒ → G, then there exists k ∈ N such that ϕ(b) = uk(a1, · · · , an) and ϕ(c) = vk(a1, · · · , an). Clearly rk(b, c) / ∈ R and so rk(uk(a1, · · · , an), vk(a1, · · · , an)) = 1. But since rk(b, c) / ∈ R, it follows that rk(b, c) = 1 in H, which is a contradiction.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Enumeration reducibility

Definition

If A, B ⊆ N, then A is enumeration reducible to B, written A ≤e B, if there exists a recursively enumerable subset R ⊆ N × Fin(N) such that n ∈ A ⇐ ⇒ (n, F) ∈ R for some finite subset F ⊆ B.

Remark

Intuitively, A ≤e B if there is a fixed effective procedure which produces an enumeration of A from any enumeration of B.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Enumeration reducibility

Definition

If A, B ⊆ N, then A is enumeration reducible to B, written A ≤e B, if there exists a recursively enumerable subset R ⊆ N × Fin(N) such that n ∈ A ⇐ ⇒ (n, F) ∈ R for some finite subset F ⊆ B.

Remark

Intuitively, A ≤e B if there is a fixed effective procedure which produces an enumeration of A from any enumeration of B.

Observation

It is easily checked that if A ≤1 B, then A ≤e B.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Enumeration reducibility

Definition

If A, B ⊆ N, then A is enumeration reducible to B, written A ≤e B, if there exists a recursively enumerable subset R ⊆ N × Fin(N) such that n ∈ A ⇐ ⇒ (n, F) ∈ R for some finite subset F ⊆ B.

Remark

Intuitively, A ≤e B if there is a fixed effective procedure which produces an enumeration of A from any enumeration of B.

Remark

If G, H ∈ Gfg and G ֒ → H, then Rel(G) ≤e Rel(H) and Nonrel(G) ≤e Nonrel(H).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Enumeration reducibility

Definition

≡e is the countable Borel equivalence relation on 2N defined by A ≡e B ⇐ ⇒ A ≤e B and B ≤e A.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Enumeration reducibility

Definition

≡e is the countable Borel equivalence relation on 2N defined by A ≡e B ⇐ ⇒ A ≤e B and B ≤e A.

Theorem (Folklore)

≡T is Borel reducible to ≡e.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Enumeration reducibility

Definition

≡e is the countable Borel equivalence relation on 2N defined by A ≡e B ⇐ ⇒ A ≤e B and B ≤e A.

Theorem (Folklore)

≡T is Borel reducible to ≡e.

Proof.

A ≤T B ⇐ ⇒ A ⊕ (N A) ≤e B ⊕ (N B).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal f.g. groups revisited

Definition

If A ∈ 2N, then the finitely generated group G is relatively universal

• f e-degree A if Sk(G) = { H ∈ Gfg | Rel(H) ≤e A }.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal f.g. groups revisited

Definition

If A ∈ 2N, then the finitely generated group G is relatively universal

• f e-degree A if Sk(G) = { H ∈ Gfg | Rel(H) ≤e A }.

Observation

If A, B ∈ 2N and GA, GB are relatively universal of e-degrees A, B, then GA ≈ GB ⇐ ⇒ A ≡e B.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Relatively universal f.g. groups revisited

Definition

If A ∈ 2N, then the finitely generated group G is relatively universal

• f e-degree A if Sk(G) = { H ∈ Gfg | Rel(H) ≤e A }.

Observation

If A, B ∈ 2N and GA, GB are relatively universal of e-degrees A, B, then GA ≈ GB ⇐ ⇒ A ≡e B.

Theorem (Higman-Scott 1998)

For each A ∈ 2N, there exists a relatively universal group GA

• f e-degree A.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Two contrasting results

Theorem

There exists a Borel map A → GA from 2N to Gfg such that: Rel(GA) ≡ e A, and if A ≡e B, then GA ≈ GB.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Two contrasting results

Theorem

There exists a Borel map A → GA from 2N to Gfg such that: Rel(GA) ≡ e A, and if A ≡e B, then GA ≈ GB.

Theorem (Thomas 2010)

If A → GA is a Borel map from 2N to Gfg such that Rel(GA) ≡ T A, then there exists a Turing degree d0 such that for all Turing degrees d ≥ T d0, there exists an infinite subset { An | n ∈ N } ⊆ d such that the groups { GAn | n ∈ N } are pairwise incomparable with respect to embeddability.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The bi-embeddability relation for Kazhdan groups

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The bi-embeddability relation for Kazhdan groups

Theorem (Thomas-Williams 2013)

For each A ∈ 2N, there exists a finitely generated group KA such that KA is relatively universal of e-degree A; and KA is a Kazhdan group.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The bi-embeddability relation for Kazhdan groups

Theorem (Thomas-Williams 2013)

For each A ∈ 2N, there exists a finitely generated group KA such that KA is relatively universal of e-degree A; and KA is a Kazhdan group.

Corollary

The bi-embeddability relation for Kazhdan groups is weakly universal.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let GA = s1, · · · , sn be relatively universal of e-degree A.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let GA = s1, · · · , sn be relatively universal of e-degree A. Let H be an infinite hyperbolic Kazhdan group.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let GA = s1, · · · , sn be relatively universal of e-degree A. Let H be an infinite hyperbolic Kazhdan group. Then H has a finite presentation H = x1, · · · , xm | T .

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let GA = s1, · · · , sn be relatively universal of e-degree A. Let H be an infinite hyperbolic Kazhdan group. Then H has a finite presentation H = x1, · · · , xm | T . By Ol’shanskii, H has a free subgroup F = a1, · · · , an such that for every N F, there exists M H such that N = M ∩ F.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let GA = s1, · · · , sn be relatively universal of e-degree A. Let H be an infinite hyperbolic Kazhdan group. Then H has a finite presentation H = x1, · · · , xm | T . By Ol’shanskii, H has a free subgroup F = a1, · · · , an such that for every N F, there exists M H such that N = M ∩ F. Let ai = ui(¯ x) for 1 ≤ i ≤ n.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Sketch proof of Theorem

Let GA = s1, · · · , sn be relatively universal of e-degree A. Let H be an infinite hyperbolic Kazhdan group. Then H has a finite presentation H = x1, · · · , xm | T . By Ol’shanskii, H has a free subgroup F = a1, · · · , an such that for every N F, there exists M H such that N = M ∩ F. Let ai = ui(¯ x) for 1 ≤ i ≤ n. Let KA = x1, · · · , xm | R , where R = T ∪ { w( u1(¯ x), · · · , un(¯ x) ) | w(a1, · · · , an) ∈ Rel(GA) }.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

How about f.g. simple groups?

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

How about f.g. simple groups?

Definition

A finitely generated group G ∈ Gfg is recursively presented if Rel(G) is recursively enumerable.

Theorem (Kuznetsov 1958)

If G is a recursively presented simple group, then G has a solvable word problem.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

How about f.g. simple groups?

Definition

A finitely generated group G ∈ Gfg is recursively presented if Rel(G) is recursively enumerable.

Theorem (Kuznetsov 1958)

If G is a recursively presented simple group, then G has a solvable word problem.

Extended Kuznetsov Theorem

If G ∈ Gfg is simple, then Nonrel(G) ≤e Rel(G).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

How about f.g. simple groups?

Definition

A finitely generated group G ∈ Gfg is recursively presented if Rel(G) is recursively enumerable.

Theorem (Kuznetsov 1958)

If G is a recursively presented simple group, then G has a solvable word problem.

Extended Kuznetsov Theorem

If G ∈ Gfg is simple, then Nonrel(G) ≤e Rel(G).

Theorem (Thomas-Williams 2013)

If A ∈ 2N and GA is relatively universal of e-degree A, then GA is not simple.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Let G = a1, · · · , an be a finitely generated simple group.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Let G = a1, · · · , an be a finitely generated simple group. Let Fn be the free group on x1, · · · , xn and let W be the set of all (not necessarily reduced) words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Let G = a1, · · · , an be a finitely generated simple group. Let Fn be the free group on x1, · · · , xn and let W be the set of all (not necessarily reduced) words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

Fix some word u ∈ W such that u(a1, · · · , an) = 1 in G.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Let G = a1, · · · , an be a finitely generated simple group. Let Fn be the free group on x1, · · · , xn and let W be the set of all (not necessarily reduced) words in x1, · · · , xn, x−1

1 , · · · , x−1 n .

Fix some word u ∈ W such that u(a1, · · · , an) = 1 in G. Since G is simple, if w ∈ W, then the following are equivalent:

(i) w ∈ Nonrel(G). (ii) The group Gw with presentation x1, · · · , xn | Rel G ∪ { w } is trivial. (iii) u is in the normal closure of Rel(G) ∪ { w } in Fn.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Consider the set R ⊆ W × Fin(W) defined by (w, F) ∈ R ⇐ ⇒ u is in the normal closure of F ∪ { w } in Fn.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Consider the set R ⊆ W × Fin(W) defined by (w, F) ∈ R ⇐ ⇒ u is in the normal closure of F ∪ { w } in Fn. Then R is recursively enumerable and the following are equivalent:

(i) w ∈ Nonrel(G). (ii) (w, F) ∈ R for some finite subset F ⊆ Rel(G).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of the Extended Kuznetsov Theorem

Consider the set R ⊆ W × Fin(W) defined by (w, F) ∈ R ⇐ ⇒ u is in the normal closure of F ∪ { w } in Fn. Then R is recursively enumerable and the following are equivalent:

(i) w ∈ Nonrel(G). (ii) (w, F) ∈ R for some finite subset F ⊆ Rel(G).

Thus Nonrel(G) ≤e Rel(G).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

A characterization of the total enumeration degrees

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

A characterization of the total enumeration degrees

Definition

A set A ∈ 2N is total if A ≡e A ⊕ (N A).

Example

If G ∈ Gfg is simple, then Rel(G) is total.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

A characterization of the total enumeration degrees

Definition

A set A ∈ 2N is total if A ≡e A ⊕ (N A).

Example

If G ∈ Gfg is simple, then Rel(G) is total.

Definition

An enumeration degree e is total if it contains a total set.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

A characterization of the total enumeration degrees

Definition

A set A ∈ 2N is total if A ≡e A ⊕ (N A).

Example

If G ∈ Gfg is simple, then Rel(G) is total.

Definition

An enumeration degree e is total if it contains a total set.

Theorem (Thomas 2016)

An enumeration degree e is total if and only if there exists a finitely generated simple group G such that Rel(G) ∈ e.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The obvious conjectures ...

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The obvious conjectures ...

Conjecture

The isomorphism relation for finitely generated simple groups is not weakly universal.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The obvious conjectures ...

Conjecture

The isomorphism relation for finitely generated simple groups is not weakly universal.

Conjecture

There does not exist an isomorphism-invariant Borel map φ : Gfg → Gfg such that for all G ∈ Gfg, φ(G) is a finitely generated simple group; and G ֒ → φ(G).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The obvious conjectures ...

Conjecture

The isomorphism relation for finitely generated simple groups is not weakly universal.

Conjecture

There does not exist an isomorphism-invariant Borel map φ : Gfg → Gfg such that for all G ∈ Gfg, φ(G) is a finitely generated Kazhdan group; and G ֒ → φ(G).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

The obvious conjectures ...

Conjecture

The isomorphism relation for finitely generated simple groups is not weakly universal.

Theorem

There exists an isomorphism-invariant Borel map φ : Gfg → Gfg such that for all G ∈ Gfg, φ(G) is a 2-generator group; and G ֒ → φ(G).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

On the other hand ...

Notation

G is the space of arbitrary countable groups.

Theorem (Ramsey cardinal)

Suppose that G → KG is any Borel map from G to Gfg such that G ֒ → KG for all G ∈ G. Then there exists an uncountable Borel family F ⊆ G of pairwise isomorphic groups such that the groups { KG | G ∈ F } are pairwise incomparable with respect to relative constructibility; i.e., if G = H ∈ F, then KG / ∈ L[ KH ] and KH / ∈ L[ KG ].

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

On the other hand ...

Notation

G is the space of arbitrary countable groups.

Theorem (Ramsey cardinal)

Suppose that G → KG is any Borel map from G to Gfg such that G ֒ → KG for all G ∈ G. Then there exists an uncountable Borel family F ⊆ G of pairwise isomorphic groups such that the groups { KG | G ∈ F } are pairwise incomparable with respect to relative constructibility; i.e., if G = H ∈ F, then KG / ∈ L[ KH ] and KH / ∈ L[ KG ].

Remark

In ZFC, we can find an uncountable Borel family F such that the groups { KG | G ∈ F } are pairwise incomparable with respect to embeddability.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Definition

An infinite group G is said to be just infinite if every proper quotient

• f G is finite.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Definition

An infinite group G is said to be just infinite if every proper quotient

• f G is finite.

Some Examples

Infinite simple groups are just infinite.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Definition

An infinite group G is said to be just infinite if every proper quotient

• f G is finite.

Some Examples

Infinite simple groups are just infinite. SL3(Z) is just infinite.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Definition

An infinite group G is said to be just infinite if every proper quotient

• f G is finite.

Some Examples

Infinite simple groups are just infinite. SL3(Z) is just infinite.

Remark

An interesting theory of just infinite groups has been developed by Girgorchuk, Wilson, etc.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Proposition (Grigorchuk 2000)

Every infinite f.g. group G has a just infinite quotient G/N.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Proposition (Grigorchuk 2000)

Every infinite f.g. group G has a just infinite quotient G/N.

Proof.

It is enough to show that the partially ordered set N = { N G | G/N is infinite } has a maximal element.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Proposition (Grigorchuk 2000)

Every infinite f.g. group G has a just infinite quotient G/N.

Proof.

It is enough to show that the partially ordered set N = { N G | G/N is infinite } has a maximal element. Suppose that N0 · · · Nℓ · · · is a chain and let N = Nℓ.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Proposition (Grigorchuk 2000)

Every infinite f.g. group G has a just infinite quotient G/N.

Proof.

It is enough to show that the partially ordered set N = { N G | G/N is infinite } has a maximal element. Suppose that N0 · · · Nℓ · · · is a chain and let N = Nℓ. If N / ∈ N, then [ G : N ] < ∞ and this implies that N is f.g., which is a contradiction.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Theorem (Thomas 2013)

There does not exist a Borel map G → QG from Gfg to Gfg such that for all G, H ∈ Gfg, QG is a just infinite quotient of G; and if G ∼ = H, then QG ∼ = QH.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Just infinite groups

Theorem (Thomas 2013)

There does not exist a Borel map G → QG from Gfg to Gfg such that for all G, H ∈ Gfg, QG is a just infinite quotient of G; and if G ∼ = H, then QG ∼ = QH. This is an easy consequence of:

Theorem (Thomas 2013)

The isomorphism relation on the space of finitely generated simple groups is not smooth.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

A nonselection result

Notation

If x ∈ 2N, then ¯ x(n) = 1 − x(n).

Definition

E∗

0 is the Borel equivalence relation on 2N defined by

x E∗

0 y

⇐ ⇒ x E0 y or x E0 ¯ y.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

A nonselection result

Notation

If x ∈ 2N, then ¯ x(n) = 1 − x(n).

Definition

E∗

0 is the Borel equivalence relation on 2N defined by

x E∗

0 y

⇐ ⇒ x E0 y or x E0 ¯ y.

Proposition (Folklore)

There does not exist a Borel map θ : 2N → 2N such that: if x E∗

0 y, then θ(x) E0 θ(y);

θ(x) E∗

0 x.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Proposition

Suppose that the Borel map θ : 2N → 2N selects an E0-class within each E∗

0-class.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Proposition

Suppose that the Borel map θ : 2N → 2N selects an E0-class within each E∗

0-class.

Let µ be the usual product probability measure on 2N and let X = { x ∈ 2N | x E0 y for some y ∈ θ[ 2N] }.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Proposition

Suppose that the Borel map θ : 2N → 2N selects an E0-class within each E∗

0-class.

Let µ be the usual product probability measure on 2N and let X = { x ∈ 2N | x E0 y for some y ∈ θ[ 2N] }. Then X is a Borel tail event and hence µ(X) = 0, 1.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Proposition

Suppose that the Borel map θ : 2N → 2N selects an E0-class within each E∗

0-class.

Let µ be the usual product probability measure on 2N and let X = { x ∈ 2N | x E0 y for some y ∈ θ[ 2N] }. Then X is a Borel tail event and hence µ(X) = 0, 1. Since the map x → ¯ x is measure preserving, it follows that µ(2N X) = µ(X), which is impossible.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Theorem

Suppose ϕ : Gfg → Gfg is a Borel map such that

ϕ(G) is a just infinite quotient of G; and if G ∼ = H, then ϕ(G) ∼ = ϕ(H).

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Theorem

Suppose ϕ : Gfg → Gfg is a Borel map such that

ϕ(G) is a just infinite quotient of G; and if G ∼ = H, then ϕ(G) ∼ = ϕ(H).

Let z → Sz a Borel reduction from E0 to the isomorphism relation ∼ = on infinite f.g. simple groups.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Theorem

Suppose ϕ : Gfg → Gfg is a Borel map such that

ϕ(G) is a just infinite quotient of G; and if G ∼ = H, then ϕ(G) ∼ = ϕ(H).

Let z → Sz a Borel reduction from E0 to the isomorphism relation ∼ = on infinite f.g. simple groups. Define ψ : 2N → Gfg by z

ψ

→ Gz = Sz × S¯

z.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Theorem

Suppose ϕ : Gfg → Gfg is a Borel map such that

ϕ(G) is a just infinite quotient of G; and if G ∼ = H, then ϕ(G) ∼ = ϕ(H).

Let z → Sz a Borel reduction from E0 to the isomorphism relation ∼ = on infinite f.g. simple groups. Define ψ : 2N → Gfg by z

ψ

→ Gz = Sz × S¯

z.

Each ϕ(Gz) is isomorphic to either Sz or S¯

z.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Theorem

Suppose ϕ : Gfg → Gfg is a Borel map such that

ϕ(G) is a just infinite quotient of G; and if G ∼ = H, then ϕ(G) ∼ = ϕ(H).

Let z → Sz a Borel reduction from E0 to the isomorphism relation ∼ = on infinite f.g. simple groups. Define ψ : 2N → Gfg by z

ψ

→ Gz = Sz × S¯

z.

Each ϕ(Gz) is isomorphic to either Sz or S¯

z.

Thus the Borel map θ : 2N → 2N defined by θ(z) = y ⇐ ⇒ y ∈ { z, ¯ z } and ϕ(Gz) ∼ = Sy selects an E0-class within each E∗

0-class, which is a contradiction.

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016

Proof of Theorem

Suppose ϕ : Gfg → Gfg is a Borel map such that

ϕ(G) is a just infinite quotient of G; and if G ∼ = H, then ϕ(G) ∼ = ϕ(H).

Let z → Sz a Borel reduction from E0 to the isomorphism relation ∼ = on infinite f.g. simple groups. Define ψ : 2N → Gfg by z

ψ

→ Gz = Sz × S¯

z.

Each ϕ(Gz) is isomorphic to either Sz or S¯

z.

Thus the Borel map θ : 2N → 2N defined by θ(z) = y ⇐ ⇒ y ∈ { z, ¯ z } and ϕ(Gz) ∼ = Sy selects an E0-class within each E∗

0-class, which is a contradiction.

The End

Simon Thomas (Rutgers University) Turin Logic Seminar 1st June 2016