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OutlineDefinitions and examplesDefinitionComputationExamplesSelf-similar groups and virtual endomorphismsSelf-similar groupsIterated Monodromy GroupsLecture 1Volodymyr NekrashevychAugust, 2009BathV. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 1 / 17OutlineOutline1Main definition.2Computation of iterated monodromy groups.3Virtual endomorphisms and self-similar groups.4Limit spaces.5Examples and applications.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 2 / 17Definitions and examples DefinitionDefinitionLet p : M1−→ M be a covering of a space by a subset (a partialself-covering).V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 3 / 17Definitions and examples DefinitionDouble self-covering of the circleConsider the map p : x 7→ 2x of the circle R/Z.The fundamental group of the circle is generated by the loop γ equal tothe image of [0, 1] in R/Z.The lifts of γ by pnare the images ofm2n,m+12n , for m = 0, . . . , 2n− 1.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 4 / 17Definitions and examples ComputationEncoding of the treeChoose an alphabet X, |X| = deg p, a bijection Λ : X → p−1(t), and a pathℓ(x) from t to Λ(x) for every x ∈ X.Define the map Λ : X∗→ T inductively by the rule:Λ(xv) is the end of the p|v |-lift of ℓ(x) starting at Λ(v ).The map Λ : X∗→ T is an isomorphism of rooted trees, where v isconnected to vy in X∗.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 5 / 17Definitions and examples ComputationRecurrent formulaLet us identify the trees T and X∗using the isomorphism Λ. Then theiterated monodromy group acts on the tree X∗. Let γ be an element ofthe fundamental group π1(M, t).PropositionFor x ∈ X, let γxbe the lift of γ by p starting at Λ(x). Let y ∈ X be suchthat Λ (y ) is the end of γx.Then for every v ∈ X∗we haveγ(xv) = yℓ(x)γxℓ(y)−1(v).V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 6 / 17Definitions and examples ComputationΛ(v)Λ(u)p−n(ℓ(x))p−n(ℓ(y))p−n(γx)Λ(xv)Λ(yu)V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 7 / 17Definitions and examples ExamplesExample: z2p : z 7→ z2induces a double self-covering of C \ {0} (homotopicallyequivalent to the 2-fold self-covering of the circle).Chose the basepoint t = 1. p−1(1) = {1, −1}. Let ℓ(0 ) be trivial, and letℓ(1) be the unit upper half-circle. Let γ be the unit circle based at t withthe positiv e orientation.t = 1−1ℓ(1)γ0γ1We get γ(0 v ) = 1v, γ(1v ) = 0γ(v ). This is known as the addingmachine.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 8 / 17Definitions and examples ExamplesExample: −z32+3z2A rati onal function f (z) ∈ C(z) is post-critically finite if orbit of everycritical point of f is fini te. The union Pfof the orbits of critical values isthe post-critical set of f .If f is post-critically finite, then it is a partial self-covering ofbC \ Pf.Consider f (z) = −z32+3z2. It has three critical points ∞ , 1, −1, which arefixed under f .Hence it is post-critically finite and is a covering of C \{±1} by the subsetC \ f−1({±1}) = C \ {±1, ±2}.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 9 / 17Definitions and examples ExamplesExample: −z32+3z2Let t = 0. It has three preimages 0, ±√3. Choose t he followingconnecting paths and generators of π1(C \ {±1}, 0) (ℓ(0 ) is trivial):01ℓ(1)ℓ(2)−1abV. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 10 / 17Definitions and examples ExamplesExample: −z32+3z2The generators a and b are lift ed to the following paths:01ℓ(1)ℓ(2)−1f−1(a)f−1(a)f−1(b)f−1(b)−22a(0v ) = 1v, a(1v ) = 0a(v ), a(2v) = 2 v ,b(0v ) = 2 v , b(1 v ) = 1v, b(2v ) = 0 b(v).V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 11 / 17Definitions and examples ExamplesA multi-dimensional exampleConsider the map F of C2:(x, y ) 7→1 −y2x2, 1 −1x2It can be naturally extended to the projective plane.(x : y : z) 7→ (x2− y2: x2− z2: x2).The set {x = 0} ∪ {y = 0} ∪ {z = 0} i s the critical locus. Thepost-critical set is the union of the line at infinity with t he lin esx = 0, x = 1, y = 0, y = 1, x = y .They are permuted as follows:{x = 0} 7→ {z = 0} 7→ {y = 1} 7→ {x = y} 7→ {x = 0}{y = 0} 7→ {x = 1} 7→ {y = 0}.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 12 / 17Definitions and examples ExamplesThe iterated monodromy group of F (as computed by J. Belk andS. Koch) is generated by the transformations:a(1v ) = 1b(v ), a(2v ) = 2v , a(3v) = 3v , a(4v ) = 4b(v),b(1v) = 1c(v ), b(2v) = 2c(v ), b(3v) = 3v, b(4v ) = 4v ,c(1v) = 4d(v), c(2v ) = 3(ceb)−1(v), c(3v ) = 2(fa)−1(v), c(4v ) = 1v,d(1v) = 2v, d(2v ) = 1a(v ), d(3v) = 4v , d(4v) = 3a(v ),e(1v ) = 1f (v ), e(2v ) = 2v , e(3v ) = 3f (v), e(4v) = 4v,f (1v) = 3b−1(v), f (2v ) = 4v, f (3v) = 1eb(v), f (4v) = 2e(v).V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 13 / 17Self-similar groups and virtual endomorphisms Self-similar groupsWe have seen that for every g ∈ IMG (p) and for every x ∈ X there existsy ∈ X and gx∈ IMG (p) such thatg(xv) = ygx(v)for all v ∈ X∗.Groups satisfying this condition are called self-similar.The map πg: x 7→ y is a permutation (describing the action of g on thefirst level of the tree. Hence we get a mapg 7→ πg(g1, g2, . . . , gd),from IMG (p) to Sd≀ IMG (p), where X = {1, 2, . . . , d }. It is easy tocheck that this map is a homomorphism.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 14 / 17Self-similar groups and virtual endomorphisms Self-similar groupsDefinitionA wreath recursion on a group G is a homomorphismΦ : G −→ Sd≀ G .The wreath defining IMG (p) depends on the choice of the bijection of Xwith p−1(t) and on the choice of the connecting paths ℓ(x). D ifferentchoices produce wreath recursions, which differ from each other byapplication of an inner automorphism of Sd≀ G .We say that Φ1, Φ2: G −→ Sd≀ G are equivalent if there exists an innerautomorphism τ of Sd≀ G such that Φ2= τ ◦ Φ1.V. Nekrashevych (Texas A&M) Iterated monodromy groups August, 2009 Bath 15 / 17Self-similar groups and virtual endomorphisms Self-similar groupsEvery wreath recursion defines an action on the tree