Quasi-Isometric Diversity of Marked Groups

Quasi-isometric diversity of marked groups

A. Minasyan, D. Osin, S. Witzel

March 10, 2020

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups S and T are quasi-isometric, written S ∼q.i. T , if there ∃ a q.i. S → T .

Proposition

∼q.i. is an equivalence relation on the class of metric spaces.

For G = hX i, dX denotes the word metric. Example. 1 If G is a ﬁnitely generated group and X , Y are ﬁnite generating sets of G, then (G, dX ) ∼q.i. (G, dY ).

2 (G, dX ) ∼q.i Cay(G, X ).

Quasi-isometry between metric spaces

Let (S, dS ), (T , dT ) be metric spaces.

Deﬁnition (Gromov) A map f : S → T is a quasi-isometry if ∃ λ, c, ε such that 1 (a) λ dT (f (x), f (y)) − c ≤ dS (x, y) ≤ λdT (f (x), f (y)) + c ∀ x, y ∈ S; (b) f (S) is ε-dense in T .

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Proposition

∼q.i. is an equivalence relation on the class of metric spaces.

For G = hX i, dX denotes the word metric. Example. 1 If G is a ﬁnitely generated group and X , Y are ﬁnite generating sets of G, then (G, dX ) ∼q.i. (G, dY ).

2 (G, dX ) ∼q.i Cay(G, X ).

Quasi-isometry between metric spaces

Let (S, dS ), (T , dT ) be metric spaces.

S and T are quasi-isometric, written S ∼q.i. T , if there ∃ a q.i. S → T .

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups For G = hX i, dX denotes the word metric. Example. 1 If G is a ﬁnitely generated group and X , Y are ﬁnite generating sets of G, then (G, dX ) ∼q.i. (G, dY ).

2 (G, dX ) ∼q.i Cay(G, X ).

Quasi-isometry between metric spaces

Let (S, dS ), (T , dT ) be metric spaces.

S and T are quasi-isometric, written S ∼q.i. T , if there ∃ a q.i. S → T .

Proposition

∼q.i. is an equivalence relation on the class of metric spaces.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Example. 1 If G is a ﬁnitely generated group and X , Y are ﬁnite generating sets of G, then (G, dX ) ∼q.i. (G, dY ).

2 (G, dX ) ∼q.i Cay(G, X ).

Quasi-isometry between metric spaces

Let (S, dS ), (T , dT ) be metric spaces.

S and T are quasi-isometric, written S ∼q.i. T , if there ∃ a q.i. S → T .

Proposition

∼q.i. is an equivalence relation on the class of metric spaces.

For G = hX i, dX denotes the word metric.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups 2 (G, dX ) ∼q.i Cay(G, X ).

Quasi-isometry between metric spaces

Let (S, dS ), (T , dT ) be metric spaces.

S and T are quasi-isometric, written S ∼q.i. T , if there ∃ a q.i. S → T .

Proposition

∼q.i. is an equivalence relation on the class of metric spaces.

For G = hX i, dX denotes the word metric. Example. 1 If G is a ﬁnitely generated group and X , Y are ﬁnite generating sets of G, then (G, dX ) ∼q.i. (G, dY ).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Quasi-isometry between metric spaces

Let (S, dS ), (T , dT ) be metric spaces.

S and T are quasi-isometric, written S ∼q.i. T , if there ∃ a q.i. S → T .

Proposition

∼q.i. is an equivalence relation on the class of metric spaces.

For G = hX i, dX denotes the word metric. Example. 1 If G is a ﬁnitely generated group and X , Y are ﬁnite generating sets of G, then (G, dX ) ∼q.i. (G, dY ).

2 (G, dX ) ∼q.i Cay(G, X ).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Examples.

1 If M is a compact manifold, then π1(M) ∼q.i. Me.

2 T

2 2 Cay(Z ⊕ Z, X } ⊂ R = Te

2 If G is ﬁnitely generated and H ≤ G is of ﬁnite index, then G ∼q.i. H. (Hint: consider H y Cay(G, X )).

Examples

Theorem (Svarc-Milnor Lemma) If S is a geodesic metric space and G y S isometrically, properly, and cocompactly, then G is ﬁnitely generated and G ∼q.i. S.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups 2 T

2 2 Cay(Z ⊕ Z, X } ⊂ R = Te

2 If G is ﬁnitely generated and H ≤ G is of ﬁnite index, then G ∼q.i. H. (Hint: consider H y Cay(G, X )).

Examples

Theorem (Svarc-Milnor Lemma) If S is a geodesic metric space and G y S isometrically, properly, and cocompactly, then G is ﬁnitely generated and G ∼q.i. S.

Examples.

1 If M is a compact manifold, then π1(M) ∼q.i. Me.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups 2 If G is ﬁnitely generated and H ≤ G is of ﬁnite index, then G ∼q.i. H. (Hint: consider H y Cay(G, X )).

Examples

Theorem (Svarc-Milnor Lemma) If S is a geodesic metric space and G y S isometrically, properly, and cocompactly, then G is ﬁnitely generated and G ∼q.i. S.

Examples.

1 If M is a compact manifold, then π1(M) ∼q.i. Me.

2 T

2 2 Cay(Z ⊕ Z, X } ⊂ R = Te

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Examples

Theorem (Svarc-Milnor Lemma) If S is a geodesic metric space and G y S isometrically, properly, and cocompactly, then G is ﬁnitely generated and G ∼q.i. S.

Examples.

1 If M is a compact manifold, then π1(M) ∼q.i. Me.

2 T

2 2 Cay(Z ⊕ Z, X } ⊂ R = Te

2 If G is ﬁnitely generated and H ≤ G is of ﬁnite index, then G ∼q.i. H. (Hint: consider H y Cay(G, X )).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Theorem (Grigorchuk, 1984) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated groups.

Invariant: growth functions.

Theorem (Bowditch, 1998) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated C 0(1/6) groups.

Invariant: growth of tout loops.

Theorem (Cornulier-Tessera, 2013) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated solvable groups.

Invariant: The set of ultraﬁlters corresponding to simply connected asymptotic cones.

How many quasi-isometry classes are there?

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Invariant: growth functions.

Theorem (Bowditch, 1998) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated C 0(1/6) groups.

Invariant: growth of tout loops.

Theorem (Cornulier-Tessera, 2013) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated solvable groups.

Invariant: The set of ultraﬁlters corresponding to simply connected asymptotic cones.

How many quasi-isometry classes are there?

Theorem (Grigorchuk, 1984) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated groups.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Theorem (Bowditch, 1998) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated C 0(1/6) groups.

Invariant: growth of tout loops.

Theorem (Cornulier-Tessera, 2013) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated solvable groups.

Invariant: The set of ultraﬁlters corresponding to simply connected asymptotic cones.

How many quasi-isometry classes are there?

Theorem (Grigorchuk, 1984) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated groups.

Invariant: growth functions.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Invariant: growth of tout loops.

Theorem (Cornulier-Tessera, 2013) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated solvable groups.

Invariant: The set of ultraﬁlters corresponding to simply connected asymptotic cones.

How many quasi-isometry classes are there?

Theorem (Grigorchuk, 1984) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated groups.

Invariant: growth functions.

Theorem (Bowditch, 1998) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated C 0(1/6) groups.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Theorem (Cornulier-Tessera, 2013) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated solvable groups.

Invariant: The set of ultraﬁlters corresponding to simply connected asymptotic cones.

How many quasi-isometry classes are there?

Theorem (Grigorchuk, 1984) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated groups.

Invariant: growth functions.

Theorem (Bowditch, 1998) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated C 0(1/6) groups.

Invariant: growth of tout loops.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Invariant: The set of ultraﬁlters corresponding to simply connected asymptotic cones.

How many quasi-isometry classes are there?

Theorem (Grigorchuk, 1984) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated groups.

Invariant: growth functions.

Theorem (Bowditch, 1998) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated C 0(1/6) groups.

Invariant: growth of tout loops.

Theorem (Cornulier-Tessera, 2013) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated solvable groups.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups How many quasi-isometry classes are there?

Theorem (Grigorchuk, 1984) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated groups.

Invariant: growth functions.

Theorem (Bowditch, 1998) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated C 0(1/6) groups.

Invariant: growth of tout loops.

Theorem (Cornulier-Tessera, 2013) There exist 2ℵ0 quasi-isometry classes of ﬁnitely generated solvable groups.

Invariant: The set of ultraﬁlters corresponding to simply connected asymptotic cones.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Let G = hAi, H = hBi, where A = (a1,..., an), B = (b1,..., bn).

—(G, A) and (H, B) are k-similar, denoted (G, A) ≈k (H, B), if there is an isomorphism between balls of radius k in Cay(G, A) and Cay(H, B) mapping edges labelled by ai to edges labelled by bi , i = 1,..., n. —(G, A) and (H, B) are equivalent, denoted (G, A) ≈ (H, B), if (G, A) ≈k (H, B) ∀ k; equivalently, ai 7→ bi extends to an isomorphism G → H.

Deﬁnition (The Grigorchuk space)

Gn = {(G, (a1,..., an)) | G = ha1,..., ani}/ ≈

lim (Gi , Ai ) → (G, A) if ∀ k (Gi , Ai ) ≈k (G, A) eventually holds. i→∞

Theorem

For all n, Gn is a totally disconnected, compact, metrizable space.

The Grigorchuck space

An n-generated marked group is a pair (G, A), where G is a group generated by an ordered n-tuple A ⊆ G.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups —(G, A) and (H, B) are equivalent, denoted (G, A) ≈ (H, B), if (G, A) ≈k (H, B) ∀ k; equivalently, ai 7→ bi extends to an isomorphism G → H.

Deﬁnition (The Grigorchuk space)

Gn = {(G, (a1,..., an)) | G = ha1,..., ani}/ ≈

lim (Gi , Ai ) → (G, A) if ∀ k (Gi , Ai ) ≈k (G, A) eventually holds. i→∞

Theorem

For all n, Gn is a totally disconnected, compact, metrizable space.

The Grigorchuck space

An n-generated marked group is a pair (G, A), where G is a group generated by an ordered n-tuple A ⊆ G. Let G = hAi, H = hBi, where A = (a1,..., an), B = (b1,..., bn).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Deﬁnition (The Grigorchuk space)

Gn = {(G, (a1,..., an)) | G = ha1,..., ani}/ ≈

lim (Gi , Ai ) → (G, A) if ∀ k (Gi , Ai ) ≈k (G, A) eventually holds. i→∞

Theorem

For all n, Gn is a totally disconnected, compact, metrizable space.

The Grigorchuck space

An n-generated marked group is a pair (G, A), where G is a group generated by an ordered n-tuple A ⊆ G. Let G = hAi, H = hBi, where A = (a1,..., an), B = (b1,..., bn).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups lim (Gi , Ai ) → (G, A) if ∀ k (Gi , Ai ) ≈k (G, A) eventually holds. i→∞

Theorem

For all n, Gn is a totally disconnected, compact, metrizable space.

The Grigorchuck space

An n-generated marked group is a pair (G, A), where G is a group generated by an ordered n-tuple A ⊆ G. Let G = hAi, H = hBi, where A = (a1,..., an), B = (b1,..., bn).

Deﬁnition (The Grigorchuk space)

Gn = {(G, (a1,..., an)) | G = ha1,..., ani}/ ≈

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Theorem

For all n, Gn is a totally disconnected, compact, metrizable space.

The Grigorchuck space

An n-generated marked group is a pair (G, A), where G is a group generated by an ordered n-tuple A ⊆ G. Let G = hAi, H = hBi, where A = (a1,..., an), B = (b1,..., bn).

Deﬁnition (The Grigorchuk space)

Gn = {(G, (a1,..., an)) | G = ha1,..., ani}/ ≈

lim (Gi , Ai ) → (G, A) if ∀ k (Gi , Ai ) ≈k (G, A) eventually holds. i→∞

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups The Grigorchuck space

An n-generated marked group is a pair (G, A), where G is a group generated by an ordered n-tuple A ⊆ G. Let G = hAi, H = hBi, where A = (a1,..., an), B = (b1,..., bn).

Deﬁnition (The Grigorchuk space)

Gn = {(G, (a1,..., an)) | G = ha1,..., ani}/ ≈

lim (Gi , Ai ) → (G, A) if ∀ k (Gi , Ai ) ≈k (G, A) eventually holds. i→∞

Theorem

For all n, Gn is a totally disconnected, compact, metrizable space.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Fn N (Fn) = {N C Fn} ⊆ 2 .

Given (G, A) ∈ Gn, where A = (a1,..., an), there is

ε(G,A) : Fn → G

such that ε(G,A)(xi ) = ai ∀ i.

Proposition

The map (G, A) 7→ Ker ε(G,A) deﬁnes a homeomorphism Gn → N (Fn).

Examples.

1 limi→∞(Z/mZ, {1}) = (Z, {1}). ∞ ∞ 2 S T If N = i=1 Ni or N = i=1 Ni , then limi→∞ Ni = N in N (Fn).

In particular, the set of finitely presented groups is dense in Gn and every residually finite group is a limit of finite groups.

Equivalent deﬁnition (the Chabauty space)

Fn Let Fn = F (x1,..., xn) be the free group of rank n. Endow 2 with the product topology.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Given (G, A) ∈ Gn, where A = (a1,..., an), there is

ε(G,A) : Fn → G

such that ε(G,A)(xi ) = ai ∀ i.

Proposition

The map (G, A) 7→ Ker ε(G,A) deﬁnes a homeomorphism Gn → N (Fn).

Examples.

1 limi→∞(Z/mZ, {1}) = (Z, {1}). ∞ ∞ 2 S T If N = i=1 Ni or N = i=1 Ni , then limi→∞ Ni = N in N (Fn).

In particular, the set of finitely presented groups is dense in Gn and every residually finite group is a limit of finite groups.

Equivalent deﬁnition (the Chabauty space)

Fn Let Fn = F (x1,..., xn) be the free group of rank n. Endow 2 with the product topology. Fn N (Fn) = {N C Fn} ⊆ 2 .

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Proposition

The map (G, A) 7→ Ker ε(G,A) deﬁnes a homeomorphism Gn → N (Fn).

Examples.

1 limi→∞(Z/mZ, {1}) = (Z, {1}). ∞ ∞ 2 S T If N = i=1 Ni or N = i=1 Ni , then limi→∞ Ni = N in N (Fn).

In particular, the set of finitely presented groups is dense in Gn and every residually finite group is a limit of finite groups.

Equivalent deﬁnition (the Chabauty space)

Fn Let Fn = F (x1,..., xn) be the free group of rank n. Endow 2 with the product topology. Fn N (Fn) = {N C Fn} ⊆ 2 .

Given (G, A) ∈ Gn, where A = (a1,..., an), there is

ε(G,A) : Fn → G

such that ε(G,A)(xi ) = ai ∀ i.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups ∞ ∞ 2 S T If N = i=1 Ni or N = i=1 Ni , then limi→∞ Ni = N in N (Fn).

In particular, the set of finitely presented groups is dense in Gn and every residually finite group is a limit of finite groups.

Equivalent deﬁnition (the Chabauty space)

Fn Let Fn = F (x1,..., xn) be the free group of rank n. Endow 2 with the product topology. Fn N (Fn) = {N C Fn} ⊆ 2 .

Given (G, A) ∈ Gn, where A = (a1,..., an), there is

ε(G,A) : Fn → G

such that ε(G,A)(xi ) = ai ∀ i.

Proposition

The map (G, A) 7→ Ker ε(G,A) deﬁnes a homeomorphism Gn → N (Fn).

Examples.

1 limi→∞(Z/mZ, {1}) = (Z, {1}).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups In particular, the set of finitely presented groups is dense in Gn and every residually finite group is a limit of finite groups.

Equivalent deﬁnition (the Chabauty space)

Fn Let Fn = F (x1,..., xn) be the free group of rank n. Endow 2 with the product topology. Fn N (Fn) = {N C Fn} ⊆ 2 .

Given (G, A) ∈ Gn, where A = (a1,..., an), there is

ε(G,A) : Fn → G

such that ε(G,A)(xi ) = ai ∀ i.

Proposition

The map (G, A) 7→ Ker ε(G,A) deﬁnes a homeomorphism Gn → N (Fn).

Examples.

1 limi→∞(Z/mZ, {1}) = (Z, {1}). ∞ ∞ 2 S T If N = i=1 Ni or N = i=1 Ni , then limi→∞ Ni = N in N (Fn).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Equivalent deﬁnition (the Chabauty space)

Fn Let Fn = F (x1,..., xn) be the free group of rank n. Endow 2 with the product topology. Fn N (Fn) = {N C Fn} ⊆ 2 .

Given (G, A) ∈ Gn, where A = (a1,..., an), there is

ε(G,A) : Fn → G

such that ε(G,A)(xi ) = ai ∀ i.

Proposition

The map (G, A) 7→ Ker ε(G,A) deﬁnes a homeomorphism Gn → N (Fn).

Examples.

1 limi→∞(Z/mZ, {1}) = (Z, {1}). ∞ ∞ 2 S T If N = i=1 Ni or N = i=1 Ni , then limi→∞ Ni = N in N (Fn).

In particular, the set of finitely presented groups is dense in Gn and every residually finite group is a limit of finite groups.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Deﬁnition

A subset S ⊆ Gn is quasi-isometrically diverse if every comeagre subset of S has 2ℵ0 quasi-isometry classes.

Theorem (Minasyan–O.–Witzel)

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

A subset of a topological space is perfect if it is closed and has no isolated points.

Corollary

Suppose that S ⊆ Gn is perfect and contains a dense subset of ﬁnitely presented groups. Then S is quasi-isometrically diverse.

The main theorem

A subset of a topological space is meager (respectively, comeager) if it is a union of countably many nowhere dense sets (respectively, an intersection of countably many sets with dense interiors).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Theorem (Minasyan–O.–Witzel)

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

A subset of a topological space is perfect if it is closed and has no isolated points.

Corollary

Suppose that S ⊆ Gn is perfect and contains a dense subset of ﬁnitely presented groups. Then S is quasi-isometrically diverse.

The main theorem

A subset of a topological space is meager (respectively, comeager) if it is a union of countably many nowhere dense sets (respectively, an intersection of countably many sets with dense interiors).

Deﬁnition

A subset S ⊆ Gn is quasi-isometrically diverse if every comeagre subset of S has 2ℵ0 quasi-isometry classes.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups A subset of a topological space is perfect if it is closed and has no isolated points.

Corollary

Suppose that S ⊆ Gn is perfect and contains a dense subset of ﬁnitely presented groups. Then S is quasi-isometrically diverse.

The main theorem

A subset of a topological space is meager (respectively, comeager) if it is a union of countably many nowhere dense sets (respectively, an intersection of countably many sets with dense interiors).

Deﬁnition

A subset S ⊆ Gn is quasi-isometrically diverse if every comeagre subset of S has 2ℵ0 quasi-isometry classes.

Theorem (Minasyan–O.–Witzel)

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Corollary

Suppose that S ⊆ Gn is perfect and contains a dense subset of ﬁnitely presented groups. Then S is quasi-isometrically diverse.

The main theorem

A subset of a topological space is meager (respectively, comeager) if it is a union of countably many nowhere dense sets (respectively, an intersection of countably many sets with dense interiors).

Deﬁnition

A subset S ⊆ Gn is quasi-isometrically diverse if every comeagre subset of S has 2ℵ0 quasi-isometry classes.

Theorem (Minasyan–O.–Witzel)

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

A subset of a topological space is perfect if it is closed and has no isolated points.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups The main theorem

A subset of a topological space is meager (respectively, comeager) if it is a union of countably many nowhere dense sets (respectively, an intersection of countably many sets with dense interiors).

Deﬁnition

A subset S ⊆ Gn is quasi-isometrically diverse if every comeagre subset of S has 2ℵ0 quasi-isometry classes.

Theorem (Minasyan–O.–Witzel)

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

A subset of a topological space is perfect if it is closed and has no isolated points.

Corollary

Suppose that S ⊆ Gn is perfect and contains a dense subset of ﬁnitely presented groups. Then S is quasi-isometrically diverse.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of ﬁnitely generated simple (torsion, divisible, property (T), ...) groups.

Idea of the proof. Let Hn be the set of all (G, A) ∈ Gn such that G is non-elementary, hyperbolic, and has trivial ﬁnite radical. Generic groups in H are simple.

— (Olshanskii + ε) Simple groups are dense in Hn.

— Simplicity is deﬁnable by a Π2-sentence in Lω1,ω:

∞ !! _ −1 ±1 −1 ±1 ∀ a ∀ b b 6= 1 → ∃ t1,..., ∃ t` a = t1 b t1 ··· t` b t` . `=1

It follows that simple groups form a Gδ set in Gn.

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of torsion free Tarski Monsters.

A diverse zoo of monsters

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Idea of the proof. Let Hn be the set of all (G, A) ∈ Gn such that G is non-elementary, hyperbolic, and has trivial ﬁnite radical. Generic groups in H are simple.

— (Olshanskii + ε) Simple groups are dense in Hn.

— Simplicity is deﬁnable by a Π2-sentence in Lω1,ω:

∞ !! _ −1 ±1 −1 ±1 ∀ a ∀ b b 6= 1 → ∃ t1,..., ∃ t` a = t1 b t1 ··· t` b t` . `=1

It follows that simple groups form a Gδ set in Gn.

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of torsion free Tarski Monsters.

A diverse zoo of monsters

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of ﬁnitely generated simple (torsion, divisible, property (T), ...) groups.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups — (Olshanskii + ε) Simple groups are dense in Hn.

— Simplicity is deﬁnable by a Π2-sentence in Lω1,ω:

∞ !! _ −1 ±1 −1 ±1 ∀ a ∀ b b 6= 1 → ∃ t1,..., ∃ t` a = t1 b t1 ··· t` b t` . `=1

It follows that simple groups form a Gδ set in Gn.

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of torsion free Tarski Monsters.

A diverse zoo of monsters

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of ﬁnitely generated simple (torsion, divisible, property (T), ...) groups.

Idea of the proof. Let Hn be the set of all (G, A) ∈ Gn such that G is non-elementary, hyperbolic, and has trivial ﬁnite radical. Generic groups in H are simple.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups — Simplicity is deﬁnable by a Π2-sentence in Lω1,ω:

∞ !! _ −1 ±1 −1 ±1 ∀ a ∀ b b 6= 1 → ∃ t1,..., ∃ t` a = t1 b t1 ··· t` b t` . `=1

It follows that simple groups form a Gδ set in Gn.

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of torsion free Tarski Monsters.

A diverse zoo of monsters

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of ﬁnitely generated simple (torsion, divisible, property (T), ...) groups.

Idea of the proof. Let Hn be the set of all (G, A) ∈ Gn such that G is non-elementary, hyperbolic, and has trivial ﬁnite radical. Generic groups in H are simple.

— (Olshanskii + ε) Simple groups are dense in Hn.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups ∞ !! _ −1 ±1 −1 ±1 ∀ a ∀ b b 6= 1 → ∃ t1,..., ∃ t` a = t1 b t1 ··· t` b t` . `=1

It follows that simple groups form a Gδ set in Gn.

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of torsion free Tarski Monsters.

A diverse zoo of monsters

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of ﬁnitely generated simple (torsion, divisible, property (T), ...) groups.

Idea of the proof. Let Hn be the set of all (G, A) ∈ Gn such that G is non-elementary, hyperbolic, and has trivial ﬁnite radical. Generic groups in H are simple.

— (Olshanskii + ε) Simple groups are dense in Hn.

— Simplicity is deﬁnable by a Π2-sentence in Lω1,ω:

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of torsion free Tarski Monsters.

A diverse zoo of monsters

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of ﬁnitely generated simple (torsion, divisible, property (T), ...) groups.

Idea of the proof. Let Hn be the set of all (G, A) ∈ Gn such that G is non-elementary, hyperbolic, and has trivial ﬁnite radical. Generic groups in H are simple.

— (Olshanskii + ε) Simple groups are dense in Hn.

— Simplicity is deﬁnable by a Π2-sentence in Lω1,ω:

∞ !! _ −1 ±1 −1 ±1 ∀ a ∀ b b 6= 1 → ∃ t1,..., ∃ t` a = t1 b t1 ··· t` b t` . `=1

It follows that simple groups form a Gδ set in Gn.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups A diverse zoo of monsters

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of ﬁnitely generated simple (torsion, divisible, property (T), ...) groups.

Idea of the proof. Let Hn be the set of all (G, A) ∈ Gn such that G is non-elementary, hyperbolic, and has trivial ﬁnite radical. Generic groups in H are simple.

— (Olshanskii + ε) Simple groups are dense in Hn.

— Simplicity is deﬁnable by a Π2-sentence in Lω1,ω:

∞ !! _ −1 ±1 −1 ±1 ∀ a ∀ b b 6= 1 → ∃ t1,..., ∃ t` a = t1 b t1 ··· t` b t` . `=1

It follows that simple groups form a Gδ set in Gn.

Corollary (Minasyan–O.–Witzel) There are 2ℵ0 quasi-isometry classes of torsion free Tarski Monsters.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Remarks. 1 Every ﬁnitely presented group of asymptotic dimension 1 is virtually free (Fujiwara-Whyte, 2007). 2 Cornulier-Tessera groups are (nilpotent of class 3)-by-abelian, while our groups are (nilpotent of class 2)-by-abelian.

Another application

Corollary (Minasyan–O.–Witzel) There exist 2ℵ0 pairwise non-quasi-isometric finitely generated groups G splitting as ∞ {1} → Z2 → G → Z2 wr Z → {1}, (1) ∞ where Z2 is central in G. In particular, (a) there exist continuously many quasi-isometry classes of finitely generated groups of asymptotic dimension 1; (b) there exist continuously many quasi-isometry classes of finitely generated center-by-metabelian groups (i.e., groups G satisfying the identity [G 00, G] = 1).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups 2 Cornulier-Tessera groups are (nilpotent of class 3)-by-abelian, while our groups are (nilpotent of class 2)-by-abelian.

Another application

Remarks. 1 Every ﬁnitely presented group of asymptotic dimension 1 is virtually free (Fujiwara-Whyte, 2007).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Another application

Remarks. 1 Every ﬁnitely presented group of asymptotic dimension 1 is virtually free (Fujiwara-Whyte, 2007). 2 Cornulier-Tessera groups are (nilpotent of class 3)-by-abelian, while our groups are (nilpotent of class 2)-by-abelian.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Elementary equivalence preserves many geometric properties of f.g. groups: polynomial growth (Gromov); hyperbolicity (Sela, Andr´e); quasi-isometry type of polycyclic groups (Sabbagh-Wilson, Raphael).

Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

Quasi-isometry vs. elementary equivalence

G and H are elementarily equivalent if they satisfy the same sentences in the ﬁrst-order language of groups.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups hyperbolicity (Sela, Andr´e); quasi-isometry type of polycyclic groups (Sabbagh-Wilson, Raphael).

Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

Quasi-isometry vs. elementary equivalence

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups quasi-isometry type of polycyclic groups (Sabbagh-Wilson, Raphael).

Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

Quasi-isometry vs. elementary equivalence

G and H are elementarily equivalent if they satisfy the same sentences in the ﬁrst-order language of groups. Elementary equivalence preserves many geometric properties of f.g. groups: polynomial growth (Gromov); hyperbolicity (Sela, Andr´e);

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

Quasi-isometry vs. elementary equivalence

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

Quasi-isometry vs. elementary equivalence

Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

Quasi-isometry vs. elementary equivalence

Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

Quasi-isometry vs. elementary equivalence

Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Quasi-isometry vs. elementary equivalence

Problem To what extent does the ﬁrst order theory of a ﬁnitely generated group determine its quasi-isometry type?

tf Let Hn denote the set of all non-cyclic torsion-free hyperbolic groups in Gn.

Theorem (O., 2020)

tf Generic groups in Hn are elementarily equivalent.

Corollary There exist 2ℵ0 pairwise non-quasi-isometric ﬁnitely generated groups having the same elementary theory.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Lemma (Thomas) For any n, N ∈ N, the set

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}

is closed in Gn × Gn.

Idea of the proof: Fix N. Let (Gi , Xi ) → (G, X ) and (Hi , Yi ) → (H, Y ). Every pointed N-q.i. maps elements of length ` to elements of length at most N(` + N). This allows us to build an N-q.i. φ:(G, X ) → (H, Y ) as a limit of N-quasi-isometries φi :(Gi , Xi ) → (Hi , Yi ).

Pointed quasi-isometries

Let (G, X ), (H, Y ) ∈ Gn. A map φ: G → H is a pointed N-quasi-isometry if φ(1) = 1 and

φ is a quasi-isometry between (G, dX ) and (H, dY ) with parameters λ = c = ε = N.

We write (G, X ) N−q.i. (H, Y ) when such a map exists.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Idea of the proof: Fix N. Let (Gi , Xi ) → (G, X ) and (Hi , Yi ) → (H, Y ). Every pointed N-q.i. maps elements of length ` to elements of length at most N(` + N). This allows us to build an N-q.i. φ:(G, X ) → (H, Y ) as a limit of N-quasi-isometries φi :(Gi , Xi ) → (Hi , Yi ).

Pointed quasi-isometries

Let (G, X ), (H, Y ) ∈ Gn. A map φ: G → H is a pointed N-quasi-isometry if φ(1) = 1 and

φ is a quasi-isometry between (G, dX ) and (H, dY ) with parameters λ = c = ε = N.

We write (G, X ) N−q.i. (H, Y ) when such a map exists.

Lemma (Thomas) For any n, N ∈ N, the set

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}

is closed in Gn × Gn.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Every pointed N-q.i. maps elements of length ` to elements of length at most N(` + N). This allows us to build an N-q.i. φ:(G, X ) → (H, Y ) as a limit of N-quasi-isometries φi :(Gi , Xi ) → (Hi , Yi ).

Pointed quasi-isometries

Let (G, X ), (H, Y ) ∈ Gn. A map φ: G → H is a pointed N-quasi-isometry if φ(1) = 1 and

φ is a quasi-isometry between (G, dX ) and (H, dY ) with parameters λ = c = ε = N.

We write (G, X ) N−q.i. (H, Y ) when such a map exists.

Lemma (Thomas) For any n, N ∈ N, the set

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}

is closed in Gn × Gn.

Idea of the proof: Fix N. Let (Gi , Xi ) → (G, X ) and (Hi , Yi ) → (H, Y ).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups This allows us to build an N-q.i. φ:(G, X ) → (H, Y ) as a limit of N-quasi-isometries φi :(Gi , Xi ) → (Hi , Yi ).

Pointed quasi-isometries

Let (G, X ), (H, Y ) ∈ Gn. A map φ: G → H is a pointed N-quasi-isometry if φ(1) = 1 and

φ is a quasi-isometry between (G, dX ) and (H, dY ) with parameters λ = c = ε = N.

We write (G, X ) N−q.i. (H, Y ) when such a map exists.

Lemma (Thomas) For any n, N ∈ N, the set

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}

is closed in Gn × Gn.

Idea of the proof: Fix N. Let (Gi , Xi ) → (G, X ) and (Hi , Yi ) → (H, Y ). Every pointed N-q.i. maps elements of length ` to elements of length at most N(` + N).

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Pointed quasi-isometries

Let (G, X ), (H, Y ) ∈ Gn. A map φ: G → H is a pointed N-quasi-isometry if φ(1) = 1 and

φ is a quasi-isometry between (G, dX ) and (H, dY ) with parameters λ = c = ε = N.

We write (G, X ) N−q.i. (H, Y ) when such a map exists.

Lemma (Thomas) For any n, N ∈ N, the set

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}

is closed in Gn × Gn.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Proof. Recall that

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}.

By Thomas’ lemma, every QI(G,X )(N) = {(H, Y ) | (G, X ) N−q.i. (H, Y )} is closed. Note that ∞ [ ∼qi = QI (N). N=1 Since q.i. classes have empty interior in S, they are meager in S. The Baire Category Theorem implies that there are uncountably many q.i. classes in every comeager subset of S.

Theorem (The Big Gun of DST a.k.a. Silver Dichotomy) A Borel equivalence relation on a Polish space has either at most countably many or 2ℵ0 equivalence classes.

Proof of the main theorem

Theorem

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups By Thomas’ lemma, every QI(G,X )(N) = {(H, Y ) | (G, X ) N−q.i. (H, Y )} is closed. Note that ∞ [ ∼qi = QI (N). N=1 Since q.i. classes have empty interior in S, they are meager in S. The Baire Category Theorem implies that there are uncountably many q.i. classes in every comeager subset of S.

Theorem (The Big Gun of DST a.k.a. Silver Dichotomy) A Borel equivalence relation on a Polish space has either at most countably many or 2ℵ0 equivalence classes.

Proof of the main theorem

Theorem

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

Proof. Recall that

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Note that ∞ [ ∼qi = QI (N). N=1 Since q.i. classes have empty interior in S, they are meager in S. The Baire Category Theorem implies that there are uncountably many q.i. classes in every comeager subset of S.

Theorem (The Big Gun of DST a.k.a. Silver Dichotomy) A Borel equivalence relation on a Polish space has either at most countably many or 2ℵ0 equivalence classes.

Proof of the main theorem

Theorem

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

Proof. Recall that

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}.

By Thomas’ lemma, every QI(G,X )(N) = {(H, Y ) | (G, X ) N−q.i. (H, Y )} is closed.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Since q.i. classes have empty interior in S, they are meager in S. The Baire Category Theorem implies that there are uncountably many q.i. classes in every comeager subset of S.

Theorem (The Big Gun of DST a.k.a. Silver Dichotomy) A Borel equivalence relation on a Polish space has either at most countably many or 2ℵ0 equivalence classes.

Proof of the main theorem

Theorem

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

Proof. Recall that

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}.

By Thomas’ lemma, every QI(G,X )(N) = {(H, Y ) | (G, X ) N−q.i. (H, Y )} is closed. Note that ∞ [ ∼qi = QI (N). N=1

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Theorem (The Big Gun of DST a.k.a. Silver Dichotomy) A Borel equivalence relation on a Polish space has either at most countably many or 2ℵ0 equivalence classes.

Proof of the main theorem

Theorem

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

Proof. Recall that

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Proof of the main theorem

Theorem

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

Proof. Recall that

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}.

Theorem (The Big Gun of DST a.k.a. Silver Dichotomy) A Borel equivalence relation on a Polish space has either at most countably many or 2ℵ0 equivalence classes.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups Proof of the main theorem

Theorem

Let n ∈ N and let S be a non-empty closed subset of Gn. Suppose that every non-empty open subset of S contains at least two non-quasi-isometric groups. Then S is quasi-isometrically diverse.

Proof. Recall that

QI (N) = {((G, X ), (H, Y )) ∈ Gn × Gn | (G, X ) N−q.i. (H, Y )}.

Theorem (The Big Gun of DST a.k.a. Silver Dichotomy) A Borel equivalence relation on a Polish space has either at most countably many or 2ℵ0 equivalence classes.

A. Minasyan, D. Osin, S. Witzel Quasi-isometric diversity of marked groups