On l 2 -Betti numbers and their analogues in positive characteristic - PowerPoint PPT Presentation

On l 2 -Betti numbers and their analogues in positive characteristic Andrei Jaikin-Zapirain Birmingham, August 12th, 2017 L 2 -Betti numbers Andrei Jaikin-Zapirain

The initial setting G is a finitely generated group. G > G 1 > G 2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Mat n × m ( K [ G ]) φ A K [ G / G i ] n K [ G / G i ] m G / G i : → ( v 1 , . . . , v n ) �→ ( v 1 , . . . , v n ) A . dim K Im φ A dim K ker φ A G / Gi G / Gi rk G / G i ( A ) = = n − | G : G i | | G : G i | { rk G / G i } is a collection of Sylvester matrix rank functions on K [ G ]. L 2 -Betti numbers Andrei Jaikin-Zapirain

The initial setting G is a finitely generated group. G > G 1 > G 2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Mat n × m ( K [ G ]) φ A K [ G / G i ] n K [ G / G i ] m G / G i : → ( v 1 , . . . , v n ) �→ ( v 1 , . . . , v n ) A . dim K Im φ A dim K ker φ A G / Gi G / Gi rk G / G i ( A ) = = n − | G : G i | | G : G i | { rk G / G i } is a collection of Sylvester matrix rank functions on K [ G ]. L 2 -Betti numbers Andrei Jaikin-Zapirain

The initial setting G is a finitely generated group. G > G 1 > G 2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Mat n × m ( K [ G ]) φ A K [ G / G i ] n K [ G / G i ] m G / G i : → ( v 1 , . . . , v n ) �→ ( v 1 , . . . , v n ) A . dim K Im φ A dim K ker φ A G / Gi G / Gi rk G / G i ( A ) = = n − | G : G i | | G : G i | { rk G / G i } is a collection of Sylvester matrix rank functions on K [ G ]. L 2 -Betti numbers Andrei Jaikin-Zapirain

The initial setting G is a finitely generated group. G > G 1 > G 2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Mat n × m ( K [ G ]) φ A K [ G / G i ] n K [ G / G i ] m G / G i : → ( v 1 , . . . , v n ) �→ ( v 1 , . . . , v n ) A . dim K Im φ A dim K ker φ A G / Gi G / Gi rk G / G i ( A ) = = n − | G : G i | | G : G i | { rk G / G i } is a collection of Sylvester matrix rank functions on K [ G ]. L 2 -Betti numbers Andrei Jaikin-Zapirain

The initial setting G is a finitely generated group. G > G 1 > G 2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Mat n × m ( K [ G ]) φ A K [ G / G i ] n K [ G / G i ] m G / G i : → ( v 1 , . . . , v n ) �→ ( v 1 , . . . , v n ) A . dim K Im φ A dim K ker φ A G / Gi G / Gi rk G / G i ( A ) = = n − | G : G i | | G : G i | { rk G / G i } is a collection of Sylvester matrix rank functions on K [ G ]. L 2 -Betti numbers Andrei Jaikin-Zapirain

The initial setting G is a finitely generated group. G > G 1 > G 2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Mat n × m ( K [ G ]) φ A K [ G / G i ] n K [ G / G i ] m G / G i : → ( v 1 , . . . , v n ) �→ ( v 1 , . . . , v n ) A . dim K Im φ A dim K ker φ A G / Gi G / Gi rk G / G i ( A ) = = n − | G : G i | | G : G i | { rk G / G i } is a collection of Sylvester matrix rank functions on K [ G ]. L 2 -Betti numbers Andrei Jaikin-Zapirain

The initial setting G is a finitely generated group. G > G 1 > G 2 > . . . is a chain of normal subgroups of finite index with trivial intersection. In this setting G is residually finite. K is a filed (of arbitrary characteristic), A ∈ Mat n × m ( K [ G ]) φ A K [ G / G i ] n K [ G / G i ] m G / G i : → ( v 1 , . . . , v n ) �→ ( v 1 , . . . , v n ) A . dim K Im φ A dim K ker φ A G / Gi G / Gi rk G / G i ( A ) = = n − | G : G i | | G : G i | { rk G / G i } is a collection of Sylvester matrix rank functions on K [ G ]. L 2 -Betti numbers Andrei Jaikin-Zapirain

Sylvester rank function on a K -algebra Let R be a K -algebra. A Sylvester matrix rank function rk on R is a map rk : Mat ( R ) → R ≥ 0 satisfying the following conditions (SRF1) rk ( M ) = 0 if M is any zero matrix and rk (1 R ) = 1; (SRF2) rk ( M 1 M 2 ) ≤ min { rk ( M 1 ) , rk ( M 2 ) } if M 1 and M 2 can be multiplied; � M 1 � 0 (SRF3) rk = rk ( M 1 ) + rk ( M 2 ); 0 M 2 � M 1 � M 3 (SRF4) rk ≥ rk ( M 1 ) + rk ( M 2 ) if M 1 , M 2 and 0 M 2 M 3 are of appropriate sizes. The space P ( R ) of Sylvester rank functions on R is a compact convex subset of R Mat ( R ) L 2 -Betti numbers Andrei Jaikin-Zapirain

Sylvester rank function on a K -algebra Let R be a K -algebra. A Sylvester matrix rank function rk on R is a map rk : Mat ( R ) → R ≥ 0 satisfying the following conditions (SRF1) rk ( M ) = 0 if M is any zero matrix and rk (1 R ) = 1; (SRF2) rk ( M 1 M 2 ) ≤ min { rk ( M 1 ) , rk ( M 2 ) } if M 1 and M 2 can be multiplied; � M 1 � 0 (SRF3) rk = rk ( M 1 ) + rk ( M 2 ); 0 M 2 � M 1 � M 3 (SRF4) rk ≥ rk ( M 1 ) + rk ( M 2 ) if M 1 , M 2 and 0 M 2 M 3 are of appropriate sizes. The space P ( R ) of Sylvester rank functions on R is a compact convex subset of R Mat ( R ) L 2 -Betti numbers Andrei Jaikin-Zapirain

Sylvester rank function on a K -algebra Let R be a K -algebra. A Sylvester matrix rank function rk on R is a map rk : Mat ( R ) → R ≥ 0 satisfying the following conditions (SRF1) rk ( M ) = 0 if M is any zero matrix and rk (1 R ) = 1; (SRF2) rk ( M 1 M 2 ) ≤ min { rk ( M 1 ) , rk ( M 2 ) } if M 1 and M 2 can be multiplied; � M 1 � 0 (SRF3) rk = rk ( M 1 ) + rk ( M 2 ); 0 M 2 � M 1 � M 3 (SRF4) rk ≥ rk ( M 1 ) + rk ( M 2 ) if M 1 , M 2 and 0 M 2 M 3 are of appropriate sizes. The space P ( R ) of Sylvester rank functions on R is a compact convex subset of R Mat ( R ) L 2 -Betti numbers Andrei Jaikin-Zapirain

Sylvester rank function on a K -algebra Let R be a K -algebra. A Sylvester matrix rank function rk on R is a map rk : Mat ( R ) → R ≥ 0 satisfying the following conditions (SRF1) rk ( M ) = 0 if M is any zero matrix and rk (1 R ) = 1; (SRF2) rk ( M 1 M 2 ) ≤ min { rk ( M 1 ) , rk ( M 2 ) } if M 1 and M 2 can be multiplied; � M 1 � 0 (SRF3) rk = rk ( M 1 ) + rk ( M 2 ); 0 M 2 � M 1 � M 3 (SRF4) rk ≥ rk ( M 1 ) + rk ( M 2 ) if M 1 , M 2 and 0 M 2 M 3 are of appropriate sizes. The space P ( R ) of Sylvester rank functions on R is a compact convex subset of R Mat ( R ) L 2 -Betti numbers Andrei Jaikin-Zapirain

Sylvester rank function on a K -algebra Let R be a K -algebra. A Sylvester matrix rank function rk on R is a map rk : Mat ( R ) → R ≥ 0 satisfying the following conditions (SRF1) rk ( M ) = 0 if M is any zero matrix and rk (1 R ) = 1; (SRF2) rk ( M 1 M 2 ) ≤ min { rk ( M 1 ) , rk ( M 2 ) } if M 1 and M 2 can be multiplied; � M 1 � 0 (SRF3) rk = rk ( M 1 ) + rk ( M 2 ); 0 M 2 � M 1 � M 3 (SRF4) rk ≥ rk ( M 1 ) + rk ( M 2 ) if M 1 , M 2 and 0 M 2 M 3 are of appropriate sizes. The space P ( R ) of Sylvester rank functions on R is a compact convex subset of R Mat ( R ) L 2 -Betti numbers Andrei Jaikin-Zapirain

Sylvester rank function on a K -algebra Let R be a K -algebra. A Sylvester matrix rank function rk on R is a map rk : Mat ( R ) → R ≥ 0 satisfying the following conditions (SRF1) rk ( M ) = 0 if M is any zero matrix and rk (1 R ) = 1; (SRF2) rk ( M 1 M 2 ) ≤ min { rk ( M 1 ) , rk ( M 2 ) } if M 1 and M 2 can be multiplied; � M 1 � 0 (SRF3) rk = rk ( M 1 ) + rk ( M 2 ); 0 M 2 � M 1 � M 3 (SRF4) rk ≥ rk ( M 1 ) + rk ( M 2 ) if M 1 , M 2 and 0 M 2 M 3 are of appropriate sizes. The space P ( R ) of Sylvester rank functions on R is a compact convex subset of R Mat ( R ) L 2 -Betti numbers Andrei Jaikin-Zapirain

The main questions Main questiones 1 Is there the limit lim i →∞ rk G / G i ( A )? 2 If the limit exists, how does it depend on the chain { G i } ? 3 What is the range of possible values for lim i →∞ rk G / G i ( A ) for a given group G ? Conjectures 1 Yes, the limit exists. 2 It does not depend on the chain { G i } . 3 Assume that there exists an upper bound for the orders of finite subgroups of G . Let lcm( G ) = lcm {| H | : H is a finite subgroup of G } . 1 lcm( G ) Z . Then lim i →∞ rk G / G i ( A ) ∈ L 2 -Betti numbers Andrei Jaikin-Zapirain

The main questions Main questiones 1 Is there the limit lim i →∞ rk G / G i ( A )? 2 If the limit exists, how does it depend on the chain { G i } ? 3 What is the range of possible values for lim i →∞ rk G / G i ( A ) for a given group G ? Conjectures 1 Yes, the limit exists. 2 It does not depend on the chain { G i } . 3 Assume that there exists an upper bound for the orders of finite subgroups of G . Let lcm( G ) = lcm {| H | : H is a finite subgroup of G } . 1 lcm( G ) Z . Then lim i →∞ rk G / G i ( A ) ∈ L 2 -Betti numbers Andrei Jaikin-Zapirain