On J ø rgensen, Gehring — Martin — Tan and Tan numbers for groups of ҥgure-eight orbifolds Alexander Masley (joint work with Andrei Vesnin) Sobolev Institute of Mathematics and Laboratory of Quantum Topology Second China-Russia Workshop on Knot Theory and Related Topics Novosibirsk, August, 21, 2015 Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Classiҥcation of elements and Discrete subgroup of PSL(2 , C ) Let (︃ a )︃ b | a | 2 + | b | 2 + | c | 2 + | d | 2 . √︁ M = ∈ SL(2 , C ) , tr( M ) = a + d , ‖ M ‖ = c d A matrix M ∈ SL(2 , C ) , such that M ̸ = ± I , is called tr 2 ( M ) ∈ [0 , 4) , - elliptic if tr 2 ( M ) = 4 , - parabolic if tr 2 ( M ) / - loxodromic if ∈ [0 , 4] . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Classiҥcation of elements and Discrete subgroup of PSL(2 , C ) Let (︃ a )︃ b | a | 2 + | b | 2 + | c | 2 + | d | 2 . √︁ M = ∈ SL(2 , C ) , tr( M ) = a + d , ‖ M ‖ = c d A matrix M ∈ SL(2 , C ) , such that M ̸ = ± I , is called tr 2 ( M ) ∈ [0 , 4) , - elliptic if tr 2 ( M ) = 4 , - parabolic if tr 2 ( M ) / - loxodromic if ∈ [0 , 4] . Denote PSL(2 , C ) = SL(2 , C ) / {± I } . Deҥnition An element g ∈ PSL(2 , C ) is called elliptic , parabolic , or loxodromic if so is its representative in SL(2 , C ) . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Classiҥcation of elements and Discrete subgroup of PSL(2 , C ) Let (︃ a )︃ b | a | 2 + | b | 2 + | c | 2 + | d | 2 . √︁ M = ∈ SL(2 , C ) , tr( M ) = a + d , ‖ M ‖ = c d A matrix M ∈ SL(2 , C ) , such that M ̸ = ± I , is called tr 2 ( M ) ∈ [0 , 4) , - elliptic if tr 2 ( M ) = 4 , - parabolic if tr 2 ( M ) / - loxodromic if ∈ [0 , 4] . Denote PSL(2 , C ) = SL(2 , C ) / {± I } . Deҥnition An element g ∈ PSL(2 , C ) is called elliptic , parabolic , or loxodromic if so is its representative in SL(2 , C ) . Consider PSL(2 , C ) with the quotient topology of the matrix norm ‖ · ‖ . Deҥnition A subgroup G of PSL(2 , C ) is said to be discrete if G is a discrete set. Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Jorgensen numbers and extreme groups Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Elementary subgroups of PSL(2 , C ) and J ø rgensen inequality Let H 3 be the Poincar´ e half-space model of the hyperbolic 3-space, i. e. with the metric ds 2 = ( | dz | 2 + dt 2 ) / t 2 . ⃒ z = x + yi ∈ C , t > 0 {︁ ⃒ }︁ the set ( z , t ) Identify ∂ H 3 with C . The group PSL(2 , C ) acts on H 3 as the group of all orientation-preserving isometries and on C as the group of all linear fractional transformations. Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Elementary subgroups of PSL(2 , C ) and J ø rgensen inequality Let H 3 be the Poincar´ e half-space model of the hyperbolic 3-space, i. e. with the metric ds 2 = ( | dz | 2 + dt 2 ) / t 2 . ⃒ z = x + yi ∈ C , t > 0 {︁ ⃒ }︁ the set ( z , t ) Identify ∂ H 3 with C . The group PSL(2 , C ) acts on H 3 as the group of all orientation-preserving isometries and on C as the group of all linear fractional transformations. Deҥnition A subgroup G of PSL(2 , C ) is called elementary if there exists a finite G -orbit in H 3 ∪ C . Otherwise, it is non-elementary . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Elementary subgroups of PSL(2 , C ) and J ø rgensen inequality Let H 3 be the Poincar´ e half-space model of the hyperbolic 3-space, i. e. with the metric ds 2 = ( | dz | 2 + dt 2 ) / t 2 . ⃒ z = x + yi ∈ C , t > 0 {︁ ⃒ }︁ the set ( z , t ) Identify ∂ H 3 with C . The group PSL(2 , C ) acts on H 3 as the group of all orientation-preserving isometries and on C as the group of all linear fractional transformations. Deҥnition A subgroup G of PSL(2 , C ) is called elementary if there exists a finite G -orbit in H 3 ∪ C . Otherwise, it is non-elementary . Theorem (T. J ø rgensen, 1976) Suppose that elements f , g ∈ PSL(2 , C ) generate a non-elementary discrete ⃒ tr 2 ( f ) − 4 ⃒ + ⃒ tr[ f , g ] − 2 ⃒ ≥ 1 . ⃒ ⃒ ⃒ ⃒ group. Then the following inequality holds: For f , g ∈ PSL(2 , C ) denote ⃒ tr 2 ( f ) − 4 ⃒ + ⃒ tr[ f , g ] − 2 ⃒ ⃒ ⃒ ⃒ 𝒦 ( f , g ) = ⃒ . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
J ø rgensen number of a subgroup of PSL(2 , C ) and Extreme subgroups of PSL(2 , R ) Deҥnition Let G be a two-generated non-elementary discrete subgroup of PSL(2 , C ) . The value 𝒦 ( G ) = inf ⟨ f , g ⟩ = G 𝒦 ( f , g ) is called the J ø rgensen number of G . A two-generated discrete group G is said to be extreme if it can be generated by f and g such that 𝒦 ( f , g ) = 1 . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
J ø rgensen number of a subgroup of PSL(2 , C ) and Extreme subgroups of PSL(2 , R ) Deҥnition Let G be a two-generated non-elementary discrete subgroup of PSL(2 , C ) . The value 𝒦 ( G ) = inf ⟨ f , g ⟩ = G 𝒦 ( f , g ) is called the J ø rgensen number of G . A two-generated discrete group G is said to be extreme if it can be generated by f and g such that 𝒦 ( f , g ) = 1 . Theorem (T. J ø rgensen, M. Kiikka, 1975) The only extreme subgroups of PSL(2 , R ) are triangle groups f , g | f 2 = g 3 = ( fg ) n = I ⟨︁ ⟩︁ T (2 , 3 , n ) = , where n ≥ 7 or n = ∞ . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
J ø rgensen number of a subgroup of PSL(2 , C ) and Extreme subgroups of PSL(2 , R ) Deҥnition Let G be a two-generated non-elementary discrete subgroup of PSL(2 , C ) . The value 𝒦 ( G ) = inf ⟨ f , g ⟩ = G 𝒦 ( f , g ) is called the J ø rgensen number of G . A two-generated discrete group G is said to be extreme if it can be generated by f and g such that 𝒦 ( f , g ) = 1 . Theorem (T. J ø rgensen, M. Kiikka, 1975) The only extreme subgroups of PSL(2 , R ) are triangle groups f , g | f 2 = g 3 = ( fg ) n = I ⟨︁ ⟩︁ T (2 , 3 , n ) = , where n ≥ 7 or n = ∞ . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Collection of results about J ø rgensen numbers and extreme group Corollary The modular group PSL(2 , Z ) is extreme. Proof. It is well known, that PSL(2 , Z ) = T (2 , 3 , ∞ ) . By previous theorem, T (2 , 3 , ∞ ) is extreme. � Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Collection of results about J ø rgensen numbers and extreme group Corollary The modular group PSL(2 , Z ) is extreme. Proof. It is well known, that PSL(2 , Z ) = T (2 , 3 , ∞ ) . By previous theorem, T (2 , 3 , ∞ ) is extreme. � Theorem (H. Sato, 2000) The Picard group PSL(2 , Z [ i ]) is extreme. Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Collection of results about J ø rgensen numbers and extreme group Corollary The modular group PSL(2 , Z ) is extreme. Proof. It is well known, that PSL(2 , Z ) = T (2 , 3 , ∞ ) . By previous theorem, T (2 , 3 , ∞ ) is extreme. � Theorem (H. Sato, 2000) The Picard group PSL(2 , Z [ i ]) is extreme. na, A. Rom´ Theorem (F. Gonz´ alez ҫ Acu˜ irez, 2007) All extreme subgroups of the Picard group are listed. Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Collection of results about J ø rgensen numbers and extreme group Corollary The modular group PSL(2 , Z ) is extreme. Proof. It is well known, that PSL(2 , Z ) = T (2 , 3 , ∞ ) . By previous theorem, T (2 , 3 , ∞ ) is extreme. � Theorem (H. Sato, 2000) The Picard group PSL(2 , Z [ i ]) is extreme. na, A. Rom´ Theorem (F. Gonz´ alez ҫ Acu˜ irez, 2007) All extreme subgroups of the Picard group are listed. Theorem (M. Oichi, H. Sato, 2006) (1) Let n ∈ N . There exists the group with Jorgensen number equals n . (2) Let r ∈ R and r > 4 . There exists the group with Jorgensen number equals r . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Figure-eight knot group Let 4 1 be the figure-eight knot in the 3-sphere S 3 . The fundamental group π 1 ( S 3 ∖ 4 1 ) is called the ҥgure-eight knot group and it has the presentation π 1 ( S 3 ∖ 4 1 ) = ⟨ f , g | [ f − 1 , g ] f = g [ f − 1 , g ] ⟩ . Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Figure-eight knot group Let 4 1 be the figure-eight knot in the 3-sphere S 3 . The fundamental group π 1 ( S 3 ∖ 4 1 ) is called the ҥgure-eight knot group and it has the presentation π 1 ( S 3 ∖ 4 1 ) = ⟨ f , g | [ f − 1 , g ] f = g [ f − 1 , g ] ⟩ . It has faithful representation in PSL(2 , C ) : (︃ 1 (︃ 1 )︃ )︃ , where ω = − 1+ √− 3 1 0 f → , g → . 0 1 − ω 1 2 Alexander Masley (joint work with Andrei Vesnin) J-, GMT-,T-numbers for groups of figure-eight orbifolds
Recommend
More recommend
