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 ( x 1 , x 2 ) be a free group with basis { x 1 , x 2 } and suppose that ϕ : F ( x 1 , x 2 ) → F ( x 1 , x 2 ) is an endomorphism such that ϕ ([ x 1 , x 2 ]) = [ x 1 , x 2 ]. Ilir Snopche Asymptotic density of test elements
Nielsen’s result Let F ( x 1 , x 2 ) be a free group with basis { x 1 , x 2 } and suppose that ϕ : F ( x 1 , x 2 ) → F ( x 1 , x 2 ) is an endomorphism such that ϕ ([ x 1 , x 2 ]) = [ x 1 , x 2 ]. QUESTION: What can we say about ϕ ? Ilir Snopche Asymptotic density of test elements
Nielsen’s result Let F ( x 1 , x 2 ) be a free group with basis { x 1 , x 2 } and suppose that ϕ : F ( x 1 , x 2 ) → F ( x 1 , x 2 ) is an endomorphism such that ϕ ([ x 1 , x 2 ]) = [ x 1 , x 2 ]. 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 ( x 1 , . . . , x n ) be a free group with basis { x 1 , . . . , x n } . 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 ( x 1 , . . . , x n ) be a free group with basis { x 1 , . . . , x n } . (Zieschang, 1964) [ x 1 , x 2 ][ x 3 , x 4 ] · · · [ x 2 m − 1 , x 2 m ] is a test element of F ( x 1 , x 2 , ..., x 2 m ). 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 ( x 1 , . . . , x n ) be a free group with basis { x 1 , . . . , x n } . (Zieschang, 1964) [ x 1 , x 2 ][ x 3 , x 4 ] · · · [ x 2 m − 1 , x 2 m ] is a test element of F ( x 1 , x 2 , ..., x 2 m ). (Rips, 1981) [ x 1 , x 2 , ..., x n ] is a test element of F ( x 1 , ..., x n ). 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 ( x 1 , . . . , x n ) be a free group with basis { x 1 , . . . , x n } . (Zieschang, 1964) [ x 1 , x 2 ][ x 3 , x 4 ] · · · [ x 2 m − 1 , x 2 m ] is a test element of F ( x 1 , x 2 , ..., x 2 m ). (Rips, 1981) [ x 1 , x 2 , ..., x n ] is a test element of F ( x 1 , ..., x n ). (Zieschang, 1965) x k 1 x k 2 · · · x k n is a test element of F ( x 1 , ..., x n ) 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 x 1 , . . . , x k ∈ X ± 1 such that g = x 1 . . . x k . 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 x 1 , . . . , x k ∈ X ± 1 such that g = x 1 . . . x k . The word metric on G with respect to X is defined by d X ( g , h ) = | g − 1 h | 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 x 1 , . . . , x k ∈ X ± 1 such that g = x 1 . . . x k . The word metric on G with respect to X is defined by d X ( g , h ) = | g − 1 h | X for g , h ∈ G . We denote by B X ( r ) = { g ∈ G | d X ( e , g ) = | g | X ≤ r } the ball of radius r ≥ 0 centered at the identity in the metric space ( G , d X ). 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 | S ∩ B X ( k ) | ρ X ( S ) = lim sup . | B X ( k ) | 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 | S ∩ B X ( k ) | ρ X ( S ) = lim sup . | B X ( k ) | 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 ρ X ( S ) instead of ρ X ( 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 | S ∩ B X ( k ) | ρ X ( S ) = lim sup . | B X ( k ) | 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 ρ X ( S ) instead of ρ X ( S ). A subset S of G is generic in G (with respect to X ) if ρ X ( S ) = 1, and it is negligible if ρ X ( S ) = 0. Ilir Snopche Asymptotic density of test elements
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