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

Galois coverings of Schreier graphs

• f 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 ζY (t) of graph Y is defined as

ζY (t) =

• [C] prime cycle in Y

(1 − tν(C))−1, (1) 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, ζY (t) = (1 − tn)−2.

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Ihara-Bass determinant formula

The Ihara-Bass’s Theorem establishes the connection between ζY (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 jth diagonal entry qj such that qj + 1 = degree of jth vertex of Y and r be the rank of fundamental group of Y , r − 1 = |E| − |V |. Then Ihara determinant formula is ζY (t)−1 = (1 − t2)r−1 det(I − At + Qt2).

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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) = {x1, x2, x3, x4}. 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 = Gπ = 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

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Examples

c d b a Y d3 b2 a2 c2 d1 b4 a4 c4 d5 b6 a6 c6 a3 b3 d2 c1 a1 b1 d4 c5 a5 b5 d6 c3

• Y

6

Examples

Y

• Y

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Examples

c d b a Y d2 b3 a3 c3 d3 b2 a2 c2 d1 b1 a1 c1

• Y

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• 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 Aρ =

• g∈G

A(g) ⊗ ρ(g), each taken dρ(= 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

• f the graph Y .

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• By setting Qρ = Q ⊗ Idρ, with dρ = degree of ρ, we have the

following analogue L(t, ρ, Y |Y )−1 = (1 − t2)(r−1)dρ det(I − tAρ + t2Qρ). Thus we have zeta functions of Y factors as follows ζ

Y (t) =

• ρ∈

G

L(t, ρ, Y |Y )dρ.

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Assumptions

• Let G be a group generated by bounded automaton A with

generating set S = {s1, · · · , sm}.

• G has level transitive action on the regular rotted tree Td.
• Recall for every s ∈ S we have s = (s|x1, · · · , s|xd)ψs, where ψs ∈ Sd

and s|x = the restriction s at x where x ∈ X = {x1, · · · , xd}.

• We call ψs as root permutation associated to state s.
• Denote ΨG = group generated by root permutations ψs

ΨG = < ψs : s ∈ S > .

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Post critical sequences

Definition A left-infinite sequence · · · x2x1 over X is called post critical if there exists a left-infinite path · · · e2, e1 in the Moore diagram of A avoiding the trivial state labeled by · · · x2x1| · · · y2y1 for some yi ∈ X. G is a group generated by bounded automaton iff the set of post critical sequences say PA is finite.

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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 Γr 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.

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Example: Basilica group

The Basilica group B1: a = (b, 1)e, b = (a, 1)ψb where ψb = (0, 1) and e is the identity in S2. a b 1 1|1 1|0 0|1 0|0 0|0, 1|1 Post critical sequences: P = {(0)−ω, (10)−ω, (01)−ω}

• 1R. 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.

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Schreier graphs of Basilica group

b b−1 b b−1 a−1 a a−1 a 0

1

BΓ1 b b−1 b b−1 a a−1 a a−1 b b−1 b b−1 a−1 a a−1 a 10

00 01 11

BΓ2 b b−1 b b−1 a a−1 a a−1 b−1b b−1 b b−1 b b−1 b a a−1 a a−1 b b−1 b b−1 a−1 a a−1 a a−1 a a−1 a

110 010 000 100 101 001 011 111

BΓ3

Figure 1: The graphs BΓ1,B Γ2 and BΓ3 are the Schreier graphs of the Basilica group (B) over X, X 2 and X 3 respectively.

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Schreier graphs of Basilica group

BΓ4 16

Schreier graphs of Basilica group

BΓ5 17

Tile graphs of Basilica group

b b−1 a−1 a

1

BΓ′ 1 b b−1 a a−1 b b−1 b b−1 a−1 a a−1 a 10

00 01 11

BΓ′ 2 b b−1 b b−1 a a−1b−1 b b−1 b b−1 b a a−1 a a−1 b b−1 b b−1 a−1 a a−1 a a−1 a a−1 a

110 010 000 100 101 001 011 111

BΓ′ 3

Figure 2: The graphs BΓ′

1,B Γ′ 2 and BΓ′ 3 are the Tile graphs of the Basilica

group (B) over X, X 2 and X 3 respectively.

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Examples

Gupta-Sidki p group2: a = (b, b−1, 1, · · · , 1, a)e, b = (1, · · · , 1)ψb, where ψb = (1, · · · , p) and e ∈ Sp and P = {(p)−w, (p)−w1, (p)−w2}.

a a−1 a a−1 a a−1 b−1 b b−1 b b−1 b

1 2 3

GSΓ3 1

21 11 31 32 22 12 13 33 23

GSΓ3 2

Schreier graphs of Gupta-Sidki p = 3 group (GS)

• 2N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Mathematische

Zeitschrift, 182 (1983) 385–388.

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Examples

Brunner-Sidki-Vieira (BSV)-group3: a = (1, · · · , 1, a−1)ψa, b = (1, · · · , 1, b)ψb where ψa = ψb = (1, 2, · · · , n) ∈ Sn and P = {(4)−w, (41)−w, (14)−w}.

b b b b a a a a

1 2 3 4

BSV Γ1

Schreier graphs

• f BSV group

11 21 31 41 12 22 32 42 13 23 33 43 14 24 34 44

BSV Γ2

• 3A. Brunner, S. Sidki, and AC Vieira, A just-nonsolvable torsion-free group defined on

the binary tree, Journal of Algebra, 211 (1999) 99–114.

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Examples

Tower of Hanoi group Hn 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}. 1 2 3

TΓ3 1

Schreier graphs of Tower of hanoi group 31 21 11 22 12 32 13 33 23

TΓ3 2 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 RotΓ : X n × K → X n × K is defined by RotΓ(v, s) = (v ′, s′), for all v, v ′ ∈ X n, s, s′ ∈ K. Definition The generalized replacement product Γn g Γr 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), s ∈ S, s|u = 1, s|uv = 1. (2) Rot((v, u), s) = ((s|u(v), s(u)), s−1), s ∈ S, s|u = 1, s|uv = 1. (3)

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Proposition If n, r ≥ 1, then the following holds:

• 1. The graphs GΓn g

GΓr, GΓn+r are isomorphic.

• 2. GΓn+r is an unramified, dn sheeted graph covering of GΓr.

Proposition

• 1. The first rotation map gives the |X| disjoint copies of tile graph GΓ′

r

indexed by x ∈ X.

• 2. In addition to the first rotation map, the second rotation map adds

the edges between the copies of GΓ′

r and it produces the tile graph GΓ′ 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 GΓ′

r+1 and it produces the Schreier graph GΓr+1.

Applying the first and second rotation maps to the tile graph GΓ′

r is

identical to the construction of inflation.

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

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Examples: Generalized replacement product of graphs

1

GrΓ1

1

GrΓ′ 1

(0, 1) (0, 0) (1, 0) (1, 1)

GrΓ2 25

Examples: Generalized replacement product of graphs

a a a a b b b b

1 2 3

BSV Γ1

1 2 3

BSV Γ′ 1

(0, 0) (0, 1) (0, 2) (0, 3) (1, 0) (1, 1) (1, 2) (1, 3) (2, 0) (2, 1) (2, 2) (2, 3) (3, 0) (3, 1) (3, 2) (3, 3)

BSV Γ1 g

BSV Γ1

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Examples: Generalized replacement product of graphs

b b−1 b b−1 b b−1 a−1 a a−1 a a−1 a

1 2

GSΓ3 1 b b−1 b b−1 b b−1

1 2

GSΓ3′ 1 b b−1 b b b−1 b−1

(0, 1) (0, 0) (0, 2)

b b−1 b b b−1 b−1

(1, 2) (1, 1) (1, 0)

b b−1 b b b−1 b−1

(2, 0) (2, 2) (2, 1)

a−1 a a−1 a a−1 a a−1 a a−1 a a−1 a a−1 a a−1 a a−1 a GSΓ3 1 g

GSΓ3

1 27

Examples: Generalized replacement product of graphs

1 2

FGΓ1

1 2

FGΓ′ 1

(0, 0) (0, 2) (0, 1) (1, 1) (1, 0) (1, 2) (2, 2) (2, 1) (2, 0)

FGΓ1 g

FGΓ1

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Examples: Generalized replacement product of graphs

1 2

TΓ1

1 2

TΓ′ 1

(0, 2) (0, 1) (0, 0) (1, 1) (1, 0) (1, 2) (2, 0) (2, 2) (2, 1)

TΓ2 29

Given a group generated by bounded automaton, what can be said about the Galois coverings of the corresponding Schreier graphs?

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Theorem If |ΨG| = d, then the Schreier graph GΓn+1 is a Galois covering of GΓn with Galois group G = Gal(GΓn+1|GΓn) ≃ ΨG. In other words, If |ΨG| = d, then the root permutations ψs are the Frobenius automorphisms associated to GΓn+1 over GΓn.

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Proof Sketch

• Recall that GΓn+r is an unramified q-sheeted covering over GΓn.

Take r = 1, so we have a covering map π :G Γn+1 →G Γn of degree

• d. (Use: Generalized replacement product of Schreier graphs.)
• Look at the lifts of every non-tile edge of the graph GΓn which is of

the form es|u = {u, s(u)}, where s|u = 1. Define a map σs|u :

GΓn+1 → GΓn+1 such that

σs|u(vxi) = vs|u(xi), ∀ vxi ∈ X n+1.

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• By Self-similarity of G, we have

σ(es|u)(x) = ψs|u(x), for all x ∈ X. Therefore every such σ(es|u) is an automorphism and they are finite in number.

• Use the facts : |ΨG| = d and G has level transitive action to show

there are exactly d such automorphisms. ⇒ G = < ψs|u | s ∈ S, u ∈ X n with s|u = 1 > = ΨG.

• Every σ(es|u) is compatible with the covering map φ.

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L and zeta functions of Schreier graphs of FG

The covering Y =

FGΓ2 over the graph Y = FGΓ1 is 3-sheeted normal

• covering. In this case the Galois group is

G = < g = (1, 2, 3) | g 3 = e > ≃ Z

• 3Z. We now write all matrices

A(g), g ∈ G. A(e) =    1 1 1 2 1 1 1 2    , A(g) = A(g 2) =    1    . The Artinized adjacency matrices Aχi, where χi is an irreducible character of G. Aχ1 =    2 1 1 1 2 1 1 1 2    , Aχ2 = Aχ3 = A(e).

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Reciprocals of L functions for Y |Y are as follows 1) For Aχ1 ζY (t)−1 = L(t, Aχ1, Y |Y )−1 = (1 − t2)3(t − 1)(3t − 1)

• 3t2 − t + 1

2 2) As Aχ2 = Aχ3 L(t, Aχ2, Y |Y )−1 = L(t, Aχ3, Y |Y )−1 = (1 − t2)3 3t2 − t + 1 2 9t4 − 6t3 + t2 − 2t + 1 2 We have ζ

Y (t)−1 =

• χi∈{χ1,χ2,χ3}

L(t, Aχi, Y |Y )−1 = (1 − t2)9(t − 1)(3t − 1)

• 3t2 − t + 1

4 9t4 − 6t3 + t2 − 2t + 1 2 .

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Zeta and L functions of Schreier graphs of Basilica group

Reciprocals of L functions for Y |Y =

BΓ3|BΓ2 :

1) For A1 ζΓ2(t)−1 = L(t, A1, Y |Y )−1 = (1−t2)4(t−1)(3t−1)

• 3t2 + 1

9t4 − 2t2 + 1

• .

2) For Aσ L(t, Aσ, Y |Y )−1 = (1 − t2)4 3t2 − 2t + 1

• ×
• 27t6 − 18t5 + 3t4 − 4t3 + t2 − 2t + 1
• .

As Y |Y is normal covering, we have ζΓ3(t)−1 = L(t, A1, Y |Y )−1L(t, Aσ, Y |Y )−1 = (1 − t2)8(t − 1)(3t − 1)

• 3t2 + 1

3t2 − 2t + 1 9t4 − 2t2 + 1

• 27t6 − 18t5 + 3t4 − 4t3 + t2 − 2t + 1
• .

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References i

• D. D’Angeli, A. Donno and E. Sava-Huss, Connectedness and

isomorphism properties of the zig-zag product of graphs, J. Graph Theory 83 (2016) 120-151.

• Y. Ihara, On discrete subgroups of the two by two projective linear

group over p-adic fields, J. Math. Soc. Japan 18 (1966), 219-235.

• J. Fabrykowski and N. Gupta, On groups with sub-exponential

growth functions, J. Indian Math. Soc. (N.S.) 49 (1987), 249-256. R Grigorchuk. Burnside problem on periodic groups. Funktsional.

• Anal. i Prilozhen., 14 (1980) 53–54.
• R. I. Grigorchuk, Some topics in the dynamics of group actions on

rooted trees, Proc. Steklov Institute of Mathematics 273 (2011) 64-175.

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References ii

• R. I. Grigorchuk and A. ˙

Zuk, Self-similarity and branching in group theory, London Mathematical Society Lecture Note Series 339 (2007) 36-93.

• A. Shaikh, H. Bhate, Zeta functions of finite Schreier graphs and

their zig-zag products, J. Algebra and Applications 16 (2017) 1750151.

• A. Shaikh, D. D’Angeli, H. Bhate, D. Sheth, Galois coverings of

Schreier graphs of groups generated by bounded automata, Preprint.

• S. Sidki, Regular trees and their automorphisms, Monografias de

Matematica, 56 (IMPA, Rio de Janeiro, 1998).

• S. Sidki, E. F. Silva, A family of just-nonsolvable torsion-free groups

defined on n-ary trees, Mat. Contemp. 21 (2001) 255274.

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References iii

• V. Nekrashevych, Self-similar Groups, Mathematical Surveys and

Monographs 117 (American Mathematical Society, 2005).

• A. Terras, Zeta functions of graphs: A stroll through the garden 128

(Cambridge University Press, 2010). For slides: http://creativecommons.org/licenses/by-sa/4.0/

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Thank you very much for your attention!

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