Pseudogroups of foliations Algebraic invariants of solenoids The discriminant group Stable actions Wild actions Future work and references

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1 Wild solenoids Olga Lukina University of Illinois at Chicago Joint work with Steven Hurder March 25, / 25

2 Cantor laminations Let M be a compact connected metrizable topological space with a foliation F. A Cantor lamination The space (M, F) is a Cantor lamination, M admits a foliated atlas ϕ i : U i ( 1, 1) n Z i, where Z i is a Cantor set, 1 i ν. Note: The leaves of F are path-connected components of M. Associated to (ϕ i, U i ) there is the holonomy pseudogroup Γ = {h ik i k 1 h i1i 0 k 1, U ij U ij 1 }, where h ij are the restrictions of the transition maps to ϕ j (U i U j ) Z j. 2 / 25

3 Invariants of Cantor laminations In a foliated space (M, F), the holonomy pseudogroup Γ depends on the choice of the transverse sections Z = i Z i. Motivating problem Given a Cantor lamination (M, F) with holonomy pseudogroup (Z, Γ), find (algebraic) invariants of F, which do not depend on the choice of Z. In the rest of the talk, we restrict to the case when (Z, Γ) is equicontinuous, that is, Z admits a metric d, so that for any ɛ > 0 there is δ > 0 such that for all g, h Γ and all x, y Z with d(x, y) < δ we have d(g(x), g(y)) < ɛ. Examples of Cantor laminations with equicontinuous dynamics are solenoids. 3 / 25

4 Example in dimension 1: the Vietoris solenoid Let f i i 1 : S1 S 1 be a p i -to-1 self-covering map of a circle of length 1. Let P = (p 1, p 2,...) be an infinite sequence, p i > 1. A Vietoris solenoid is the inverse limit space Σ P = {(x i ) = (x 0, x 1, x 2,...) f i i 1(x i ) = x i 1 } i 0 S 1 with subspace topology from the Tychonoff topology on i 0 S1. Let s S 1, then the fibre F 0 = {(s, x 1, x 2,...)} is a Cantor section, transverse to the foliation by path-connected components. The fundamental group π 1 (S 1, 0) = Z acts on F 0 via lifts of paths in S 1, so the pseudogroup action (F 0, Γ) is induced by a group action (F 0, Z). 4 / 25

5 Weak solenoids Let M 0 be a closed manifold, x 0 M 0 be a point, and let G = π 1 (M 0, x 0 ) be the fundamental group. Let f i+1 i : M i+1 M i be a finite-to-one covering, f i+1 i (x i+1 ) = x i. The inverse limit space, called a weak solenoid, and defined by S P = lim {f i+1 i : M i+1 M i } = {(x 0, x 1, x 2, ) f i+1 i (x i+1 ) = x i } is a fibre bundle with Cantor fibre F 0. The fundamental group G of M 0 acts on F 0 by path-lifts. The pseudogroup action (F 0, Γ) is induced by the group action (F 0, G). 5 / 25

6 Solenoids in dimension 2 and higher I Weak solenoids were introduced by McCord 1965, and studied in - Schori 1966, - Rogers and Tollefson 1970, 1971, 1971/72. Recall that a topological space M is homogeneous if for every x, y X there exists a homeomorphism φ : M M such that φ(x) = y. If M is a Cantor lamination, every homeomorphism φ : M M is a foliated map, as it preserves path-connected components. McCord 1965 found sufficient conditions for a solenoid M to be a homogeneous foliated space. Schori 1966, Rogers and Tollefson 1970, 1971, 1971/72 found examples of non-homogeneous solenoids, and studied maps between solenoids. 6 / 25

7 Solenoids in dimension 2 and higher II Fokkink and Oversteegen 2002 found an algebraic criterion for a weak solenoid to be a homogeneous space, and constructed the first example of a weak solenoid which has simply connected path-connected components but is not a homogeneous space. Dyer, Hurder and Lukina 2016(1), 2017, 2016(2) explored an algebraic invariant of non-homogeneous solenoids, called the discriminant group. Today, I will talk about a generalization of this invariant, called the asymptotic discriminant, and how it classifies non-homogeneous solenoids. The talk is based on the recent joint work with Steven Hurder arxiv: / 25

8 Group chains: a tool Let S P = lim{f i+1 i : M i+1 M i } be a solenoid, G 0 = π 1 (M 0, x 0 ) and G i+1 = (f i+1 0 ) π 1 (M i+1, x i+1 ) G 0. Then G = G 0 G 1 G 2 is a chain of subgroups of finite index. The quotient G/G i is a finite set, the inclusions G i+1 G i induce surjective maps G/G i+1 G/G i, and there is a homeomorphism φ : F 0 G = lim {G/G i G/G i 1 } equivariant with respect to (F 0, G) and the action G G G : (h, (eg 0, g 1 G 1, g 2 G 2,...)) (hg 0, hg 1 G 1, hg 2 G 2,...) Thus (G, G) models the action (F 0, G, Φ). 8 / 25

9 Profinite group acting on F 0 Let C i = g G gg ig 1, then C i is a finite index normal subgroup of G, then C = lim {G/C i G/C i 1 } is a profinite group, acting transitively on G = F0, with the isotropy group Theorem (Dyer, Hurder, L. 2016(1)) D x = lim {G i /C i G i 1 /C i 1 } The profinite group C is isomorphic to the Ellis group of the action (F 0, G, Φ), and D x is isomorphic to the isotropy group of the Ellis group action on F 0. We call D x the discriminant group of the action. Since D x is a profinite group, it is either finite, or a Cantor group. 9 / 25

10 Algebra versus topology Let P = {f i+1 i : M i+1 M i i 0} be a presentation for S P. The truncated presentation P n = {f i+1 i : M i+1 M i i n} determines a solenoid S Pn, homeomorphic to S P. The truncated presentation corresponds to the truncated group chain {G i } i n with discriminant group D n. Main question of the talk What is the relationship between the discriminant groups D n, for n 0? Example: If a solenoid is homogeneous (for example, a Vietoris solenoid), then D n is trivial for all n 1. In general, this need not be the case. 10 / 25

11 Why the discriminant group changes Let {G i } i 0 be a group chain, and D x = lim{g i+1 /C i+1 G i /C i } be the discriminant group. Recall that C i = g i G i g 1 i. g G Let {G i } i n be a truncated group chain for n 0. Then it s maximal normal subgroup in G n is E n,i = g i G i g 1 i, and D n = lim{g i+1 /E n,i+1 G i /E n,i } g G n Thus C i E n,i E n+k,i, also D n = lim {G i+1 /E n,i+1 G i /E n,i }, and there are a surjective maps Λ n+k,n : D n D n+k. 11 / 25

12 Stable and wild group actions Definition: Stable group actions (Dyer, Hurder, L. 2016(2)) Let (X, G, Φ) be a group action with group chain {G i } i 0. Then (X, G, Φ) is stable if there exists n 0 such that for all m n the restriction Λ m,n : D n D m is an isomorphism. Otherwise, the action is said to be wild. Example: Vietoris solenoids are stable, with trivial discriminant group. 12 / 25

13 Realization theorems for stable actions Every finite group can be realized as a discriminant group of a stable equicontinuous action on a Cantor set. Theorem (Dyer, Hurder, L. 2016(2)) Given a finite group F, there exists a group chain {G i } i 0 in SL(n, Z) for n large enough, such that for any truncated chain {G i } i n the discriminant group D n = F. Every separable profinite group can be realized as a discriminant group of a stable equicontinuous action on a Cantor set. Theorem (Dyer, Hurder, L. 2016(2)) Given a separable profinite group K, there exists a finitely generated group G and a group chain {G i } i 0 G, such that for any truncated chain {G i } i n the discriminant group D n = K, and Dn D n+1 is an isomorphism. 13 / 25

14 Tail equivalence of discriminant groups Let A = {φ i : A i A i+1 } and B = {ψ i : B i B i+1 } be sequences of surjective group homomorphisms. Then A and B are tail equivalent if there are infinite subsequences A in and B jn such that the sequence A i0 B j0 A i1 B j1 is a sequence of surjective group homomorphisms. Asymptotic discriminant (Hurder and L. 2017) Let {G i } i 0 be a group chain with discriminant group D 0, and let D n be the discriminant group of the truncated chain {G i } i n. The asymptotic discriminant of the group chain {G i } i 0 is the tail equivalence class of the sequence of surjective group homomorphisms {D n D n+1 n 0}. These sequences are distinct from group chains {G i } i 0 : the inclusion G i+1 G i is not surjective. 14 / 25

15 Stability of the asymptotic discriminant A sequence A = {φ i : A i A i+1 } is constant if φ i is a group isomorphism for i 0. Asymptotically constant sequence A sequence B = {φ i : B i B i+1 } is asymptotically constant if it is tail equivalent to a constant sequence. Now the stability of the action (X, G, Φ) can be defined in terms of the asymptotic discriminant. Lemma An equicontinuous group action (X, G, Φ) with associated group chain {G i } i 0 is stable if and only if it s asymptotic discriminant D = {D n D n+1 } is tail equivalent to a constant sequence. 15 / 25

16 Wild solenoids Question Does the asymptotic discriminant really distinguish between different group actions? The answer is yes. Theorem (Hurder and L. 2017) For n 3, let G SL(n, Z) be a torsion-free subgroup of finite index. Then there exists uncountably many distinct homeomorphism types of weak solenoids which are wild, all with the same base manifold M 0 with fundamental group G. In the remaining part of the talk, I will construct a family of examples which give this theorem. 16 / 25

17 Lenstra s construction Let G be a finitely generated and residually finite group, and Ĝ be its profinite completion. Then G embeds as a dense subgroup of Ĝ. Let D be a closed subgroup of Ĝ such that g G gdg 1 = {e}. Let {U i } be a clopen neighborhood system of the identity in Ĝ, where U i is normal and of finite index, and i 0 U i = {e}. W i = DU i is a clopen subgroup of Ĝ, such that i 0 W i = D. Set G i = W i G. Proposition The group chain {G i } i 0, constructed as above, has the discriminant group D x = D. 17 / 25

18 Wild solenoids I Consider congruence subgroups of SL N (Z), N 3, Γ N (M) = Ker{SL N (Z) SL N (Z/MZ)}. For M 3, Γ N (M) is torsion-free. Let G be a finite-index torsion-free subgroup of SL N (Z). By the congruence subgroup property, G Γ N (M) for some M 3. Let Ĥ = p SL i M N(Z/p i Z), p i are distinct primes. Lemma Let Q : G Ĥ be the diagonal embedding. Then Q(G) is dense in Ĥ. 18 / 25

19 Wild solenoids II For p i M, let A pi be a non-trivial subgroup of SL N (Z/p i Z) such that ga pi g 1 = {e} g SL N (Z/p iz) (for example, A pi may be in Alt N embedded into SL N (Z/p i Z)). Let D = p A i M p i, and set G l = G U l, where U l = m i<l{e pi } SL N (Z/p i Z). i l Proposition The action (G, G) with group chain {G l } l 1 defined as above has the discriminant group D 0 = D. 19 / 25

20 Wild solenoids III For n 1, consider truncated chains {G l } l n. One can show that the discriminant of the action (G n,, G n ) is D n = A pi, p i n and so the asymptotic discriminant of the action (G, G) is the tail equivalence class of the system {D n D n+1 }. Remark For each n 1, the kernel of D n D n+1 is non-trivial and equal to A n. Thus the action (G, G) is wild. 20 / 25

21 Wild solenoids IV An uncountable number of distinct asymptotic discriminants is obtained by distinct choices of A pi, for N = 3. For a prime p i, let A 1 p i = a A 2 p i = 1 0 b 0 1 a Then A 1 p i has order p i, and A 2 p i has order p 2 i. a Z/pi Z, a, b Z/pi Z. 21 / 25

22 Wild solenoids V Let Σ = i M {1, 2}, so I = (s M, s M+1,...), and let D I = i M A si p i. Proposition Let I 1, I 2 Σ. Then the sequences of surjective homomorphisms {Dn I1 D I1 n+1 } and {DI2 n D I2 n+1 } are tail equivalent if and only if there is n > M such that for all i n we have s i = t i. This shows that there is uncountably many distinct tail equivalence classes of wild solenoids. 22 / 25

23 Work in progress 1 Conjecture: Let S P be a weak solenoid with group chain {G i } i 0, and suppose there exists a foliated embedding S P M, where M is a smooth finite-dimensional manifold with a C 2 -foliation F. Then the asymptotic discriminant class of {G i } i 0 is constant, that is, the group action on the fibre of the solenoid S P is stable. 2 In all examples of wild solenoids, constructed so far, the embedding G Homeo(F 0 ), where G is the fundamental group of the base manifold, has an infinitely generated kernel. Find an example of a wild solenoid where the kernel is finitely generated, or show that this is not possible. 23 / 25

24 References J. T. Rogers, Jr. and J.L. Tollefson, Involution on solenoidal spaces, Fund. Math., 73, (1971/72), R. Fokkink and L. Oversteegen, Homogeneous weak solenoids, Trans. A.M.S., 354(9), 2002, (1) J. Dyer, S. Hurder and O. Lukina, The discriminant invariant of Cantor group actions, Topology Appl., 208, 2016, J. Dyer, S. Hurder and O. Lukina, Growth and homogeneity of matchbox manifolds,indagationes Mathematicae, 28(1), 2017, (2) J. Dyer, S. Hurder and O. Lukina, Molino s theory for matchbox manifolds, arxiv: , 2016, to appear in Pacific J. Math. S. Hurder and O. Lukina, Wild solenoids, arxiv: , / 25