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Asymptotic density of test elements in free groups and surface groups

Ilir Snopche

Universidade Federal do Rio de Janeiro

Joint work with S. Tanushevski

Groups St Andrews 2017 in Birmingham

August, 2017

Ilir Snopche Asymptotic density of test elements

Nielsen’s result

Let F(x1, x2) be a free group with basis {x1, x2} and suppose that ϕ : F(x1, x2) → F(x1, x2) is an endomorphism such that ϕ([x1, x2]) = [x1, x2].

Ilir Snopche Asymptotic density of test elements

Nielsen’s result

Let F(x1, x2) be a free group with basis {x1, x2} and suppose that ϕ : F(x1, x2) → F(x1, x2) is an endomorphism such that ϕ([x1, x2]) = [x1, x2]. QUESTION: What can we say about ϕ?

Ilir Snopche Asymptotic density of test elements

Nielsen’s result

Let F(x1, x2) be a free group with basis {x1, x2} and suppose that ϕ : F(x1, x2) → F(x1, x2) is an endomorphism such that ϕ([x1, x2]) = [x1, x2]. QUESTION: What can we say about ϕ? (Nielsen, 1918) ϕ must be an automorphism.

Ilir Snopche Asymptotic density of test elements

Test elements: definition and examples

Definition A test element of a group G is an element g ∈ G with the following property: if ϕ(g) = g for an endomorphism ϕ : G → G , then ϕ must be an automorphism.

Ilir Snopche Asymptotic density of test elements

Test elements: definition and examples

Definition A test element of a group G is an element g ∈ G with the following property: if ϕ(g) = g for an endomorphism ϕ : G → G , then ϕ must be an automorphism. Let F(x1, . . . , xn) be a free group with basis {x1, . . . , xn}.

Ilir Snopche Asymptotic density of test elements

Test elements: definition and examples

Definition A test element of a group G is an element g ∈ G with the following property: if ϕ(g) = g for an endomorphism ϕ : G → G , then ϕ must be an automorphism. Let F(x1, . . . , xn) be a free group with basis {x1, . . . , xn}. (Zieschang, 1964) [x1, x2][x3, x4] · · · [x2m−1, x2m] is a test element of F(x1, x2, ..., x2m).

Ilir Snopche Asymptotic density of test elements

Test elements: definition and examples

Definition A test element of a group G is an element g ∈ G with the following property: if ϕ(g) = g for an endomorphism ϕ : G → G , then ϕ must be an automorphism. Let F(x1, . . . , xn) be a free group with basis {x1, . . . , xn}. (Zieschang, 1964) [x1, x2][x3, x4] · · · [x2m−1, x2m] is a test element of F(x1, x2, ..., x2m). (Rips, 1981) [x1, x2, ..., xn] is a test element of F(x1, ..., xn).

Ilir Snopche Asymptotic density of test elements

Test elements: definition and examples

Definition A test element of a group G is an element g ∈ G with the following property: if ϕ(g) = g for an endomorphism ϕ : G → G , then ϕ must be an automorphism. Let F(x1, . . . , xn) be a free group with basis {x1, . . . , xn}. (Zieschang, 1964) [x1, x2][x3, x4] · · · [x2m−1, x2m] is a test element of F(x1, x2, ..., x2m). (Rips, 1981) [x1, x2, ..., xn] is a test element of F(x1, ..., xn). (Zieschang, 1965) xk

1 xk 2 · · · xk n is a test element of F(x1, ..., xn)

whenever k ≥ 2.

Ilir Snopche Asymptotic density of test elements

The Retract theorem

Recall that a retract of a group G is a subgroup H ≤ G for which there exists an epimorphism r : G → H that restricts to the identity homomorphism on H; such epimorphism r is called retraction.

Ilir Snopche Asymptotic density of test elements

The Retract theorem

Recall that a retract of a group G is a subgroup H ≤ G for which there exists an epimorphism r : G → H that restricts to the identity homomorphism on H; such epimorphism r is called retraction. Note that, every element x ∈ G that belongs to a proper retract H of G is not a test element.

Ilir Snopche Asymptotic density of test elements

The Retract theorem

Recall that a retract of a group G is a subgroup H ≤ G for which there exists an epimorphism r : G → H that restricts to the identity homomorphism on H; such epimorphism r is called retraction. Note that, every element x ∈ G that belongs to a proper retract H of G is not a test element. Theorem (Turner, 1996) The test elements of a free group F of finite rank are exactly the elements not contained in any proper retract of F.

Ilir Snopche Asymptotic density of test elements

Examples of Turner groups

We say that a group G is a Turner group if it satisfies the Retract Theorem: an element g ∈ G is a test element of G if and only if g is not contained in any proper retract of G.

Ilir Snopche Asymptotic density of test elements

Examples of Turner groups

We say that a group G is a Turner group if it satisfies the Retract Theorem: an element g ∈ G is a test element of G if and only if g is not contained in any proper retract of G. The following are examples of Turner groups: (Turner, 1996) free groups of finite rank;

Ilir Snopche Asymptotic density of test elements

Examples of Turner groups

We say that a group G is a Turner group if it satisfies the Retract Theorem: an element g ∈ G is a test element of G if and only if g is not contained in any proper retract of G. The following are examples of Turner groups: (Turner, 1996) free groups of finite rank; (O’neill and Turner, 2000) finitely generated Fuchsian groups;

Ilir Snopche Asymptotic density of test elements

Examples of Turner groups

We say that a group G is a Turner group if it satisfies the Retract Theorem: an element g ∈ G is a test element of G if and only if g is not contained in any proper retract of G. The following are examples of Turner groups: (Turner, 1996) free groups of finite rank; (O’neill and Turner, 2000) finitely generated Fuchsian groups; (O’neill and Turner, 2000) all surface groups except the fundamental group of the Klein bottle;

Ilir Snopche Asymptotic density of test elements

Examples of Turner groups

We say that a group G is a Turner group if it satisfies the Retract Theorem: an element g ∈ G is a test element of G if and only if g is not contained in any proper retract of G. The following are examples of Turner groups: (Turner, 1996) free groups of finite rank; (O’neill and Turner, 2000) finitely generated Fuchsian groups; (O’neill and Turner, 2000) all surface groups except the fundamental group of the Klein bottle; (Groves, 2012) torsion free hyperbolic groups.

Ilir Snopche Asymptotic density of test elements

The word metric

Let G be a finitely generated group with a finite generating set X.

Ilir Snopche Asymptotic density of test elements

The word metric

Let G be a finitely generated group with a finite generating set X. Given g ∈ G, we denote by |g|X the smallest integer k ≥ 0 for which there exist x1, . . . , xk ∈ X ±1 such that g = x1 . . . xk.

Ilir Snopche Asymptotic density of test elements

The word metric

Let G be a finitely generated group with a finite generating set X. Given g ∈ G, we denote by |g|X the smallest integer k ≥ 0 for which there exist x1, . . . , xk ∈ X ±1 such that g = x1 . . . xk. The word metric on G with respect to X is defined by dX(g, h) = |g−1h|X for g, h ∈ G.

Ilir Snopche Asymptotic density of test elements

The word metric

Let G be a finitely generated group with a finite generating set X. Given g ∈ G, we denote by |g|X the smallest integer k ≥ 0 for which there exist x1, . . . , xk ∈ X ±1 such that g = x1 . . . xk. The word metric on G with respect to X is defined by dX(g, h) = |g−1h|X for g, h ∈ G. We denote by BX(r) = {g ∈ G | dX(e, g) = |g|X ≤ r} the ball of radius r ≥ 0 centered at the identity in the metric space (G, dX).

Ilir Snopche Asymptotic density of test elements

The asymptotic density

Given S ⊆ G, the asymptotic density of S in G with respect to X is defined as ρX(S) = lim sup

k→∞

|S ∩ BX(k)| |BX(k)| .

Ilir Snopche Asymptotic density of test elements

The asymptotic density

Given S ⊆ G, the asymptotic density of S in G with respect to X is defined as ρX(S) = lim sup

k→∞

|S ∩ BX(k)| |BX(k)| . If the actual limit exists, we refer to it as the strict asymptotic density of S in G with respect to X, and we write ρX(S) instead of ρX(S).

Ilir Snopche Asymptotic density of test elements

The asymptotic density

Given S ⊆ G, the asymptotic density of S in G with respect to X is defined as ρX(S) = lim sup

k→∞

|S ∩ BX(k)| |BX(k)| . If the actual limit exists, we refer to it as the strict asymptotic density of S in G with respect to X, and we write ρX(S) instead of ρX(S). A subset S of G is generic in G (with respect to X) if ρX(S) = 1, and it is negligible if ρX(S) = 0.

Ilir Snopche Asymptotic density of test elements

The asymptotic density

Given S ⊆ G, the asymptotic density of S in G with respect to X is defined as ρX(S) = lim sup

k→∞

|S ∩ BX(k)| |BX(k)| . If the actual limit exists, we refer to it as the strict asymptotic density of S in G with respect to X, and we write ρX(S) instead of ρX(S). A subset S of G is generic in G (with respect to X) if ρX(S) = 1, and it is negligible if ρX(S) = 0. Most of the subsets of free groups for which the asymptotic density has been studied and which could be defined by a natural algebraic property are either generic or negligible.

Ilir Snopche Asymptotic density of test elements

A question

Theorem (Kapovich - Rivin - Schupp - Shpilrain, 2005) Let T be the set of test elements of a free group F(x1, x2) of rank

• 2. Then T has intermediate density (different from 0 and 1). More

precisely, 4 9(1 − 6 π2 ) ≤ ρ{x1,x2}(T) ≤ 1 − 8 3π2 .

Ilir Snopche Asymptotic density of test elements

A question

Theorem (Kapovich - Rivin - Schupp - Shpilrain, 2005) Let T be the set of test elements of a free group F(x1, x2) of rank

• 2. Then T has intermediate density (different from 0 and 1). More

precisely, 4 9(1 − 6 π2 ) ≤ ρ{x1,x2}(T) ≤ 1 − 8 3π2 . Question 1 (K - R - S - S, 2005) Let F be a free group of rank n ≥ 3. Is the set of test elements of F negligible?

Ilir Snopche Asymptotic density of test elements

Nets

Let G be a finitely generated group with a finite generating set X. A subset S of G is called a C-net (0 ≤ C < ∞) with respect to X if dX(g, S) = inf{dX(g, s) | s ∈ S} ≤ C for all g ∈ G.

Ilir Snopche Asymptotic density of test elements

Nets

Let G be a finitely generated group with a finite generating set X. A subset S of G is called a C-net (0 ≤ C < ∞) with respect to X if dX(g, S) = inf{dX(g, s) | s ∈ S} ≤ C for all g ∈ G. Observation: S ⊆ G is a C-net (with respect to X) if and

• nly if there exist elements g1, . . . , gm ∈ BX(C) such that

G = Sg1 ∪ . . . ∪ Sgm.

Ilir Snopche Asymptotic density of test elements

Nets

Let G be a finitely generated group with a finite generating set X. A subset S of G is called a C-net (0 ≤ C < ∞) with respect to X if dX(g, S) = inf{dX(g, s) | s ∈ S} ≤ C for all g ∈ G. Observation: S ⊆ G is a C-net (with respect to X) if and

• nly if there exist elements g1, . . . , gm ∈ BX(C) such that

G = Sg1 ∪ . . . ∪ Sgm. Lemma Let G be a finitely generated group, X a finite generating set of G, and S ⊆ G. Suppose that G = Sg1 ∪ . . . ∪ Sgm for some g1, . . . , gm ∈ BX(C). Then ρX(S) ≥ lim inf

k→∞

|S ∩ BX(k)| |BX(k)| ≥ 1 m|BX(C)|.

Ilir Snopche Asymptotic density of test elements

The main result - Free groups

Theorem A (S - Tanushevski, 2015) Let F be a free group of rank n ≥ 2 and T be the set of test elements of F. Then:

Ilir Snopche Asymptotic density of test elements

The main result - Free groups

Theorem A (S - Tanushevski, 2015) Let F be a free group of rank n ≥ 2 and T be the set of test elements of F. Then: (a) T is a (3n − 2)-net with respect to every finite generating set

• f F.

Ilir Snopche Asymptotic density of test elements

The main result - Free groups

Theorem A (S - Tanushevski, 2015) Let F be a free group of rank n ≥ 2 and T be the set of test elements of F. Then: (a) T is a (3n − 2)-net with respect to every finite generating set

• f F.

(b) ρX(T) ≥ lim inf

k→∞

|T ∩ BX(k)| |BX(k)| ≥ 1 (2n+1(2n − 1) + 1)|BX(3n − 2)| for every generating set X of G.

Ilir Snopche Asymptotic density of test elements

The main result - Free groups

Theorem A (S - Tanushevski, 2015) Let F be a free group of rank n ≥ 2 and T be the set of test elements of F. Then: (a) T is a (3n − 2)-net with respect to every finite generating set

• f F.

(b) ρX(T) ≥ lim inf

k→∞

|T ∩ BX(k)| |BX(k)| ≥ 1 (2n+1(2n − 1) + 1)|BX(3n − 2)| for every generating set X of G. (c) If X is a basis of F, then ρX(T) ≤ 1 − 4n − 4 (2n − 1)2ζ(n).

Ilir Snopche Asymptotic density of test elements

Pro-p groups

Let p be a prime. A pro-p group is the inverse limit of an inverse system of finite p-groups.

Ilir Snopche Asymptotic density of test elements

Pro-p groups

Let p be a prime. A pro-p group is the inverse limit of an inverse system of finite p-groups. Given a group G, we denote by Gp the pro-p completion of G, i.e.,

• Gp = lim

← −

N∈N

G/N, where N = {N ✂ G | |G : N| = pk < ∞}.

Ilir Snopche Asymptotic density of test elements

Pro-p groups

Let p be a prime. A pro-p group is the inverse limit of an inverse system of finite p-groups. Given a group G, we denote by Gp the pro-p completion of G, i.e.,

• Gp = lim

← −

N∈N

G/N, where N = {N ✂ G | |G : N| = pk < ∞}. There is a natural homomorphism p : G → Gp, which sends g ∈ G to (gN) ∈ Gp.

Ilir Snopche Asymptotic density of test elements

Pro-p groups

Let p be a prime. A pro-p group is the inverse limit of an inverse system of finite p-groups. Given a group G, we denote by Gp the pro-p completion of G, i.e.,

• Gp = lim

← −

N∈N

G/N, where N = {N ✂ G | |G : N| = pk < ∞}. There is a natural homomorphism p : G → Gp, which sends g ∈ G to (gN) ∈ Gp. If G is residually finite-p, then p is an embedding and we identify p(G) with G.

Ilir Snopche Asymptotic density of test elements

Proof of Theorem A - Key ingredients

Theorem B (S. - Tanushevski, 2015) The test elements of a finitely generated pro-p group G are exactly the elements not contained in any proper retract of G (i.e., every finitely generated pro-p group is a Turner group).

Ilir Snopche Asymptotic density of test elements

Proof of Theorem A - Key ingredients

Theorem B (S. - Tanushevski, 2015) The test elements of a finitely generated pro-p group G are exactly the elements not contained in any proper retract of G (i.e., every finitely generated pro-p group is a Turner group). Theorem C (S. - Tanushevski, 2015) Let p be a prime, and let G be a finitely generated residually finite-p Turner group. If g ∈ G is a test element of Gp, then g is a test element of G.

Ilir Snopche Asymptotic density of test elements

The main result - Surface groups

Theorem D (S - Tanushevski, 2015) Let G = x1, . . . , x2n | [x1, x2] . . . [x2n−1, x2n] be an orientable surface group of genus n ≥ 2 and let T be the set of test elements

• f G. Then:

Ilir Snopche Asymptotic density of test elements

The main result - Surface groups

Theorem D (S - Tanushevski, 2015) Let G = x1, . . . , x2n | [x1, x2] . . . [x2n−1, x2n] be an orientable surface group of genus n ≥ 2 and let T be the set of test elements

• f G. Then:

(a) T is a (161n + 8 · 25n(n − 1)(16n + 1) + 33)-net with respect to every finite generating set of G.

Ilir Snopche Asymptotic density of test elements

The main result - Surface groups

Theorem D (S - Tanushevski, 2015) Let G = x1, . . . , x2n | [x1, x2] . . . [x2n−1, x2n] be an orientable surface group of genus n ≥ 2 and let T be the set of test elements

• f G. Then:

(a) T is a (161n + 8 · 25n(n − 1)(16n + 1) + 33)-net with respect to every finite generating set of G. (b) ρX(T) ≥ lim inf

k→∞

|T ∩ BX(k)| |BX(k)| ≥ 1 |BX(161n + 8 · 25n(n − 1)(16n + 1) + 33)|2 for every generating set X of G.

Ilir Snopche Asymptotic density of test elements

Thank You!