Galois coverings of Schreier graphs of groups generated by bounded - PowerPoint PPT Presentation

Galois coverings of Schreier graphs of groups generated by bounded automata Asif Shaikh (Joint work with D D’Angeli, H Bhate & D Sheth) June 29, 2018

Ihara zeta function Let Y = ( V , E ) be a connected graph and let t ∈ C , with | t | sufficiently small. Then the Ihara zeta function ζ Y ( t ) of graph Y is defined as � (1 − t ν ( C ) ) − 1 , ζ Y ( t ) = (1) [ C ] prime cycle in Y where [ C ] in Y is an equivalence class of tailless, back-trackless primitive cycles C in Y and length of C is ν ( C ) . Example: Cycle Graph Let Y be a cycle graph with n vertices. As there are only two primes, ζ Y ( t ) = (1 − t n ) − 2 . 1

Ihara-Bass determinant formula The Ihara-Bass’s Theorem establishes the connection between ζ Y ( t ) and the adjacency matrix A of the graph Y which is given as Theorem (Ihara and Bass) Let Q be the diagonal matrix with j th diagonal entry q j such that q j + 1 = degree of j th vertex of Y and r be the rank of fundamental group of Y , r − 1 = | E | − | V | . Then Ihara determinant formula is ζ Y ( t ) − 1 = (1 − t 2 ) r − 1 det( I − At + Qt 2 ) . 2

Unramified and d -sheeted coverings • All graphs are connected and undirected. • An unramified cover of a graph Y is a surjective graph homomorphism π : � Y → Y which is a local isomorphism. • The fiber π − 1 ( x ) = { x 1 , x 2 , x 3 , x 4 } . Here x ′ i s are representatives of copies of a spanning tree of Y . 3

Galois covering of a graph • The group of automorphisms of π is Aut ( π ) = { σ : � Y → � Y automorphism | π = π ◦ σ } . An automorphism σ is determined by its action on the fiber π − 1 ( x ) above any vertex x of Y . • Call π : � Y → Y (or � Y | Y ) a Galois or normal cover if Aut ( π ) acts transitively on one fiber and hence all fibers. Its Galois group is G = G π = Aut ( π ) = G ( � Y | Y ) . • If a fiber π − 1 ( x ) is a finite set, its cardinality is called the degree of π. A finite degree cover � Y | Y is Galois iff | G | = deg π. We call σ as Frobenius automorphism. 4

Examples � Y Y 5

Examples c 2 d 1 a 2 d b 4 b 2 c 1 a 1 d 2 b 1 b 3 c b a 4 a 3 d 4 � d 3 Y c 5 c 3 a 5 d 6 b 5 c 4 c 6 a Y a 6 d 5 b 6 6

Examples � Y Y 7

Examples c 3 d 3 d a 3 b 2 b 3 � c b Y a 2 d 2 c 2 c 1 a a 1 d 1 Y b 1 8

• Suppose � Y is normal covering of Y with Galois group G . The adjacency matrix of � Y can be block diagonalized where the blocks are of the form � A ρ = A ( g ) ⊗ ρ ( g ) , g ∈ G each taken d ρ (= dim irr rep ρ ) times and m × m matrix A ( g ) for g ∈ G is the matrix whose i , j entry is A ( g ) i , j = the number of edges in � Y between ( i , id ) to ( j , g ) , where id denotes the identity in G and m is the number of vertices of the graph Y . 9

• By setting Q ρ = Q ⊗ I d ρ , with d ρ = degree of ρ, we have the following analogue Y | Y ) − 1 = (1 − t 2 ) ( r − 1) d ρ det( I − tA ρ + t 2 Q ρ ) . L ( t , ρ, � Thus we have zeta functions of � Y factors as follows � L ( t , ρ, � Y | Y ) d ρ . ζ � Y ( t ) = ρ ∈ � G 10

Assumptions • Let G be a group generated by bounded automaton A with generating set S = { s 1 , · · · , s m } . • G has level transitive action on the regular rotted tree T d . • Recall for every s ∈ S we have s = ( s | x 1 , · · · , s | x d ) ψ s , where ψ s ∈ S d and s | x = the restriction s at x where x ∈ X = { x 1 , · · · , x d } . • We call ψ s as root permutation associated to state s . • Denote Ψ G = group generated by root permutations ψ s Ψ G = < ψ s : s ∈ S > . 11

Post critical sequences Definition A left-infinite sequence · · · x 2 x 1 over X is called post critical if there exists a left-infinite path · · · e 2 , e 1 in the Moore diagram of A avoiding the trivial state labeled by · · · x 2 x 1 | · · · y 2 y 1 for some y i ∈ X. G is a group generated by bounded automaton iff the set of post critical sequences say P A is finite. 12

Schreier and Tile graphs Let G be a group generated by bounded automaton A . The levels X r of the tree X ∗ are invariant under the action of the group G . Definition The Schreier graph Γ r of the action of G on X r , is a graph with vertex set X r and two vertices v and u are adjacent if and only if there exists s ∈ S such that s ( v ) = u. Definition The tile graph Γ ′ r of the action of G on X r , is a graph with vertex set X r and two vertices v and u are adjacent if and only if there exists s ∈ S such that s ( v ) = u and s | v = 1 . The tile graph is therefore a subgraph of the Schreier graph. In our case, tile graphs are always connected. 13

Example: Basilica group The Basilica group B 1 : a = ( b , 1 ) e , b = ( a , 1 ) ψ b where ψ b = (0 , 1) and e is the identity in S 2 . a 1 | 1 Post critical sequences: 0 | 0 0 | 1 0 | 0 , 1 | 1 1 P = { (0) − ω , (10) − ω , (01) − ω } 1 | 0 b 1 R. I. Grigorchuk and A. ˙ Zuk, On a torsion-free weakly branch group defined by a three state automaton, I. J. Algebra and Computation 12 (2002) 223–246. 14

Schreier graphs of Basilica group a − 1 b − 1 a − 1 a − 1 b − 1 a − 1 b − 1 a − 1 b b a b a 0 a 10 a a a b − 1 a − 1 b − 1 b b 1 b b − 1 00 01 11 B Γ 1 B Γ 2 a − 1 a 100 b − 1 b b − 1 a − 1 b − 1 a − 1 a − 1 b − 1 a − 1 b a b a b a a a a b − 1 b a − 1 b b − 1 a − 1 b − 1 b 110 010 000 001 011 111 b − 1 b 101 a a − 1 B Γ 3 Figure 1: The graphs B Γ 1 , B Γ 2 and B Γ 3 are the Schreier graphs of the Basilica group (B) over X , X 2 and X 3 respectively. 15

Schreier graphs of Basilica group B Γ 4 16

Schreier graphs of Basilica group B Γ 5 17

Tile graphs of Basilica group a − 1 a − 1 b − 1 b − 1 a − 1 b b a 10 a a a a − 1 b − 1 b 0 1 b b − 1 00 01 11 B Γ ′ B Γ ′ 1 2 a − 1 a 100 b b − 1 a − 1 b − 1 a − 1 b − 1 a − 1 b − 1 a − 1 b a b a b a a a b − 1 b − 1 a − 1 b − 1 b b 110 010 000 001 011 111 b 101 a a − 1 B Γ ′ 3 1 , B Γ ′ Figure 2: The graphs B Γ ′ 2 and B Γ ′ 3 are the Tile graphs of the Basilica group (B) over X , X 2 and X 3 respectively. 18

Examples Gupta-Sidki p group 2 : a = ( b , b − 1 , 1 , · · · , 1 , a ) e , b = ( 1 , · · · , 1 ) ψ b , where ψ b = (1 , · · · , p ) and e ∈ S p and P = { ( p ) − w , ( p ) − w 1 , ( p ) − w 2 } . 31 b − 1 a − 1 b − 1 a a b a − 1 b 1 2 GS Γ 3 1 12 23 21 11 GS Γ 3 a − 1 a 2 3 b − 1 b 32 22 13 33 Schreier graphs of Gupta-Sidki p = 3 group (GS) 2 N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Mathematische Zeitschrift , 182 (1983) 385–388. 19

Examples Brunner-Sidki-Vieira (BSV)-group 3 : a = ( 1 , · · · , 1 , a − 1 ) ψ a , b = ( 1 , · · · , 1 , b ) ψ b where ψ a = ψ b = (1 , 2 , · · · , n ) ∈ S n and P = { (4) − w , (41) − w , (14) − w } . 44 21 11 34 a 31 24 1 2 b a b 41 BSV Γ 1 14 BSV Γ 2 a b 4 3 b a 12 43 Schreier graphs 22 33 of BSV group 32 23 13 42 3 A. Brunner, S. Sidki, and AC Vieira, A just-nonsolvable torsion-free group defined on the binary tree, Journal of Algebra , 211 (1999) 99–114. 20

Examples Tower of Hanoi group H n for n = 3 a = ( 1 , 1 , a )(1 , 2) , b = ( 1 , b , 1 )(1 , 3) , c = ( c , 1 , 1 )(2 , 3) and P = { (1) − w , (2) − w , (3) − w } . 11 1 2 T Γ 3 1 31 21 3 32 23 T Γ 3 Schreier graphs of 2 Tower of hanoi group 22 12 13 33 21

Generalized replacement product of graphs If e = { v , v ′ } is an edge of the k -regular graph Γ which has color say s near v and s ′ near v ′ and if K is the set of colors K = { 1 , 2 , · · · , k } , then the rotation map Rot Γ : X n × K → X n × K is defined by for all v , v ′ ∈ X n , s , s ′ ∈ K . Rot Γ ( v , s ) = ( v ′ , s ′ ) , Definition The generalized replacement product Γ n g � Γ r is | S | -regular graph with vertex set X n + r = X n × X r , and whose edges are described by the following rotation map: Let ( v , u ) ∈ X n × X r Rot (( v , u ) , s ) = (( v , s ( u )) , s − 1 ) , if s ∈ S and s | u = 1 . (1) Rot (( v , u ) , s ) = (( s | u ( v ) , s ( u )) , s − 1 ) , (2) s ∈ S , s | u � = 1 , s | uv = 1 . Rot (( v , u ) , s ) = (( s | u ( v ) , s ( u )) , s − 1 ) , s ∈ S , s | u � = 1 , s | uv � = 1 . (3) 22

Proposition If n , r ≥ 1 , then the following holds: 1. The graphs G Γ n g � G Γ r , G Γ n + r are isomorphic. 2. G Γ n + r is an unramified, d n sheeted graph covering of G Γ r . Proposition 1. The first rotation map gives the | X | disjoint copies of tile graph G Γ ′ r indexed by x ∈ X . 2. In addition to the first rotation map, the second rotation map adds the edges between the copies of G Γ ′ r and it produces the tile graph G Γ ′ r +1 . 3. In addition to the first and second rotation maps, the third rotation map adds the edges between the post critical vertices of the tile graph G Γ ′ r +1 and it produces the Schreier graph G Γ r +1 . Applying the first and second rotation maps to the tile graph G Γ ′ r is identical to the construction of inflation. 23

Proposition Let Γ n and Γ r be Schreier graphs of the group generated by bounded automaton S. Then the first and second rotation maps of generalized replacement product Γ n g � Γ r and the n-th iteration of inflation are equivalent. 24