Chaotic group actions on manifolds

Aniruddha C. Naolekar^{∗} and Parameswaran Sankaran

Stat-Math Division, Indian Statistical Institute, 203 Barrackpore Trunk Road, Calcutta 700 035, INDIA

                 E-mail: tsmv9706@isical.ac.in

and Chennai Mathematical Institute, 92 G N Chetty Road, Chennai 600 017, INDIA

                 E-mail: sankaran@smi.ernet.in

Abstract: We construct chaotic actions of any finitely generated infinite abelian groups on even dimensional spheres, and of finite index subgroups of SL_{n}(ℤ) on tori. We also study chaotic group actions via compactly supported homeomorphisms on open manifolds.

∗Most of this work was done when the first author was a Post-Doctoral Fellow at the Chennai Mathematical Institute, Chennai.

AMS Mathematics Subject Classification : 54H20, 57S25. Keywords. Chaotic actions, topological transitivity, and residually finite groups.

1 Introduction

Cairns, Davis, Elton, Kolganova, and Perversi [2] introduced the notion of a chaotic group action as a generalization of chaotic dynamical systems (see definition below). They showed that a group G acts chaotically on a compact Hausdorff space if and only if G is residually finite. They constructed chaotic action of the infinite cyclic group on the 2-disk and observed that the special linear group SL_{n}(ℤ) acts chaotically the n-dimensional torus. In the present paper, we construct many chaotic group actions on (even dimensional) spheres and tori using the examples of Cairns et al. as building blocks. More precisely we show, among other things, that (i) any finitely generated infinite abelian groups act chaotically on even-dimensional spheres; (ii) any finite index subgroup of SL_{n}(ℤ), n ≥ 2, acts chaotically on the nkdimensional torus for any k ≥ 1. For precise statements see section 2. We also consider chaotic group actions on (connected) open manifolds where each element of the group is a compactly supported self homeomorphism. For a (self) homeomorphism f of space X, supp(f), the support of f, is defined to be the set {x ∈ X | f(x) ≠ x}. We say that a homeomorphism f of X is compactly supported if supp(f) is relatively compact, i.e., the closure of supp(f) is compact. A basic result is that any compactly supported homeomorphism of a connected open manifold is of infinite order (see [6]). The requirement that the group act chaotically (and effectively) via compactly supported homeomorphisms, leads to many interesting results of group theoretic nature. The results we obtain are by no means exhaustive. However, we do not know of a single chaotic group action via compactly supported homeomorphism on any open manifold. We conjecture that no such action can possibly exist on the Euclidean space ℝ^{n}, n ≥ 2. The strongest evidence for this conjecture is that it is true when one restricts attention to such classes of groups as solvable groups, groups with non-trivial center, groups which decompose as a direct product with one of the factors being finitely generated (see section 3). However a complete resolution of the conjecture has eluded us. (It is not hard to show that no group can act chaotically on any 1-dimensional manifold, the case of the circle having been covered in [2].) Most of our results regarding compactly supported chaotic actions on open manifolds are valid for any noncompact, locally compact space and are treated in that generality. For the convenience of the reader, we reproduce below the definition of a chaotic action. Definition:[2] Let G act effectively and continuously on a Hausdorff topological space X. The group G is said to act chaotically on X if

(i) (topological transitivity) given non-empty open sets U, V ⊆ X, there exists g ∈ G such that gU ∩ V ≠ ∅ and,

(ii) the set of periodic points of X, i.e., those points with finite G-orbits, is

dense in X. Convention: All group actions are assumed to be effective. Acknowledgements: We thank the referee for many valuable comments, and, in particular for pointing out to us the work of H.Furstenberg [5] and J.Banks [1]. We thank Professor G.Cairns for his comments and for sending us a copy of [3]. We thank the Department of Science and Technology of India for partial financial support.

2 Chaos on compact manifolds

In this section we shall construct chaotic group actions on specific manifolds such as spheres and tori. The building blocks of our constructions are the chaotic ℤ action on the 2-disk constructed in [2], and the chaotic action of the group SL_{n}(ℤ) on the n-torus. We now state the main results of this section.

Theorem 2.1 Let G = ℤ^{l} ⊕ F where l ≥ 1, and F = ℤ_{n_{1}} ⊕ ⋯ ⊕ ℤ_{n_{k}}, k ≥ 0. Then G acts chaotically on

(i) the 2d-dimensional sphere if k = l = d or if 0 ≤ k < l ≤ d (ii) the 2d-dimensional torus if t = l = d or if 1 ≤ t < l ≤ d where k = 2t or 2t − 1.

Let s_{0}, ⋯, s_{m−1} denote a set of generators of ℤ^{m}. The group ℤ_{m} acts on ℤ^{m} by cyclically permuting the generators s_{i}. We denote the semidirect product of ℤ^{m} with ℤ_{m} by H_{m}. Similarly let e_{i,j}, 0 ≤ i < m, 0 ≤ j < n denote a basis for a free abelian group of rank mn. The group ℤ_{m} ⊕ ℤ_{n} = ⟨r, s|r^{m}, s^{n}, rsr^{−1}s^{−1}⟩ acts on ℤ^{mn} where the action is defined by r.e_{i,j} ↦ e_{i+1,j}, and s.e_{i,j} = e_{i,j+1}, where e_{m,j} = e_{0,j}, e_{i,n} = e_{i,0}. Let H_{m,n} denote the semidirect product of ℤ^{mn} ⋊ (ℤ_{m} ⊕ ℤ_{n}).

Theorem 2.2 (i) The group H_{m} = ℤ^{m} ⋊ ℤ_{m} acts chaotically on the 2-sphere for any m ≥ 2. (ii) The group H_{m,n} = ℤ^{mn} ⋊ (ℤ_{m} ⊕ ℤ_{n}) acts chaotically on the 2-torus for any m, n ≥ 2.

Theorem 2.3 Let n ≥ 2, k ≥ 1 be integers. Every finite index subgroup of SL_{n}(ℤ) acts chaotically on the nk-dimensional torus.

Before we prove the main theorems of this section, we make a preliminary observation. Recall that the action of G on a space X is said to be topologically mixing if, given any two non-empty open sets U and V, there exists a g ∈ G and an integer n > 0, such that g^{k}(U) ∩ V ≠ ∅ for all k ≥ n.

Lemma 2.4 (i) Let G_{1} and G_{2} act chaotically on X_{1} and X_{2}. Then product action of G = G_{1} × G_{2} on X_{1} × X_{2} is chaotic. If x_{i} ∈ X_{i} is fixed by G_{i}, i = 1, 2, then G acts chaotically on the smash product X_{1} ∧ X_{2} = X_{1} × X_{2}/(X_{1} × x_{2} ∪ x_{1} × X_{2}). (ii) Let X be acted on by ℤ such that the action is topologically mixing, then the diagonal action of ℤ on X^{d} is topologically mixing for any d ≥ 1.

Proof:(i). Note that (y_{1}, y_{2}) ∈ X_{1} × X_{2} is a periodic point for the G action if and only if each y_{i} ∈ X_{i} is a periodic point for the action of G_{i} on X_{i}. This implies that the G-periodic points on X_{1} × X_{2} form a dense subset. Let U, V be arbitrary open sets in X_{1} × X_{2}. Let U_{i}, V_{i} be open subsets of X_{i}, i = 1, 2 such that U_{1} × U_{2} ⊂ U, V_{1} × V_{2} ⊂ V. Let g_{i} ∈ G_{i} be such that g_{i}(U_{i}) ∩ V_{i} ≠ ∅. Then g(U) ∩ V ≠ ∅ for g = (g_{1}, g_{2}) ∈ G. Thus the G action is topologically transitive. The second assertion of 2.4(i) is follows readily from the first. (ii) Let U = ∏_{1≤i≤d} U_{i}, V = ∏_{1≤i≤d} be basic open sets of X^{d}. Let f be a generator of the ℤ-action. Choose n so large that f^{k}(U_{i}) ∩ V_{i} ≠ ∅ for all k ≥ n, 1 ≤ i ≤ d. Then it is clear that f^{k}(U) ∩ V ≠ ∅ for all k ≥ n. This proves (ii). □ Remark: There are several other well-known results concerning topological transitivity and related concepts such as topological mixing, weak mixing etc. A theorem of Furstenberg [5] says that if f : X ⟶ X is the generator of a ℤ action which is weakly mixing, then the diagonal action of ℤ on X^{d}, d ≥ 1, is topologically transitive. Suppose ℤ acts on X chaotically such that the restriction of the action to each subgroup nℤ ⊂ ℤ, n ≥ 1 is transitive, then Banks [1] has shown that the action of ℤ is weakly mixing. Proof of 2.1: We first show that even dimensional spheres and even dimensional tori admit a chaotic ℤ actions. Indeed, the chaotic ℤ-action on the 2-disk D constructed by Cairns et al. [2] is topologically mixing. See Theorem A of [4]. Lemma 2.4(ii) implies that the diagonal action of ℤ on the cartesian product D^{d} is chaotic. The desired action of ℤ on the sphere S^{2d} is obtained by semi conjugation since S^{2d} is obtained by pinching the boundary of D^{d} to a point. Again the chaotic ℤ action on the 2-torus considered by Cairns et al. [2] is topologically mixing. By lemma 2.4(ii) again, we obtain a chaotic ℤ action on any even dimensional torus. Case 1: F = 0. Now, let G = ℤ^{d} act on X = (S^{2})^{d} where the action of the i-th factor ℤ of G on the i-th coordinate is the chaotic ℤ-action on S^{2} constructed in [2]. Note that the “north pole”, ∗, of the sphere S^{2} is fixed under this action. Hence the subspace Y = {z = (z_{1}, ⋯, z_{d}) | z_{i} = ∗ for some i} is stable under the action of G. Hence we obtain an action of G on the quotient space X/Y ≅ S^{2d}. The action of G remains chaotic on S^{2d} by lemma 2.4(i). Note that the “north pole” Y ∈ S^{2d} is a G-fixed point. Since the restriction of the above action to the diagonal copy Δ ≅ ℤ is the same as the chaotic ℤ action on S^{2d} constructed at the beginning of the proof, we see that if Δ ⊂ H ⊂ ℤ^{d} = G, then the restriction of G action to H is also chaotic on S^{2d}. In particular, any free abelian group of rank l, 1 ≤ l ≤ d, acts chaotically on S^{2d}. This prove theorem 2.1(i) when F = 0.

Before considering the case k ≥ 1, we let m ≥ 2, and construct a chaotic ℤ ⊕ ℤ_{m} action on the unit disk D in ℝ^{2}. We then use this to construct ℤ ⊕ ℤ_{m} action on the S^{2}, which is thought of the space obtained from D by collapsing its boundary, ∂D, to a point. The ℤ_{m}-action on D is generated by 2π/m rotation, r, about the center of the disk. Let D_{0} denote a sector of the disk having angle 2π/m at the center of the disk. Denote by D_{j} the image of D_{0} under the map r^{j}, 1 ≤ j < m. Our generator g of the infinite cyclic group is the homeomorphism of D which is f on D_{0} and, on D_{j}, it is the map r^{j}fr^{−j}, where f is the chaotic homeomorphism of the 2-disk constructed by Cairns, et al. [2]. Note that since the map f is identity on the boundary of D_{0}, g is a well-defined selfhomeomorphism of D. Note that rgr^{−1} is the same as the map g. Thus we have obtained an action of ℤ ⊕ ℤ_{m} on D. Now the boundary ∂D of D is stable under the action of ℤ ⊕ ℤ_{m} and hence we obtain an action of ℤ ⊕ ℤ_{m} on D/∂D = S^{2}. The resulting action is chaotic on S^{2}. Note that the “north pole”, namely the point determined by ∂D is fixed under this action. Case 2: Let F ≠ 0, so that k ≥ 1. Let G be as in 2.1(i), with k ≥ 1. Then, write G as ∏_{0≤i≤k} G_{i}, where G_{0} = ℤ^{l−k}, G_{i} = ℤ ⊕ ℤ_{n_{i}}, 1 ≤ i ≤ k. The group G_{i} acts chaotically on the sphere S_{i} := S^{2} for i ≥ 1. The group G_{0} acts chaotically on S_{0} := S^{2(d−k)} if l > k. When l = k, one has k = l = d by hypothesis. In this case, it is still true that G_{0} = {1} acts chaotically on the one point space, {∗}, again denoted S_{0}. Thus we see that G acts chaotically on the product X := ∏_{0≤i≤k} S_{i}. Note that since the north pole, ∗, of each of the spaces S_{i} are fixed by G_{i} for each i ≥ 0, we see that the subspace Y = {z = (z_{0}, ⋯, z_{k}) ⊂ X | z_{i} = ∗ for some i ≥ 0} is stable under the action of G. Hence G acts chaotically on the quotient space X/Y ≅ S^{2d}.

This completes the proof of (i). (ii) We first construct a chaotic ℤ ⊕ ℤ_{m} ⊕ ℤ_{n}-action on the 2-torus T. This is an obvious generalization of the ℤ ⊕ ℤ_{2} ⊕ ℤ_{2} action on T given in [2]. We regard T as space obtained from the unit square D = I×I with opposite sides identified in the usual manner. Now subdivide the square D into mn squares D_{i,j}, 0 ≤ i < m, 0 ≤ j < n, where D_{i,j} = [i/m, (i + 1)/m] × [j/n, (j + 1)/n]. Note that (x, y) ↦ (x + 1/m, y) and (x, y) ↦ (x, y + 1/n) define commuting homeomorphisms r, s of T of orders m and n respectively. The generator g of the infinite cyclic group acts on T as follows: Let g̃ restricted to D_{0,0} be the chaotic homeomorphism f on the 2-disk constructed by Cairns et al. [2]. g̃ restricted to D_{i,j} defined to be the composition r^{i}s^{j} ∘ f ∘ s^{−j}r^{−i}, 0 ≤ i < m, 0 ≤ j < n. It is straightforward to check that g̃ defines a homeomorphism g on T and that g commutes with r and s. Thus we obtain an action of ℤ ⊕ ℤ_{m} ⊕ ℤ_{n}. That this action is chaotic follows easily from the fact that f acts chaotically on D_{0,0}. Observe that a similar construction leads to a chaotic ℤ ⊕ ℤ_{m} action on the 2-torus T. Using these actions on 2-tori, and the chaotic action of ℤ on any even dimensional torus as building blocks constructed at the beginning of the proof, one constructs the required G action on the 2d-dimensional torus just as in the proof of (i). □ Proof of 2.2:(i). We use the notation introduced in the proof of 2.1 where we constructed a chaotic action of ℤ⊕Z_{m} on the 2-sphere. First we “expand” the ℤ ⊕ ℤ_{m} action on disk D as follows. Consider the homeomorphism s of the disk defined as s(x) = f(x), x ∈ D_{0}, and s(x) = x, x ∉ D_{0} using the notation of proof of 2.1. Let s_{j} = r^{j}sr^{−j}, 0 ≤ j ≤ m. s_{m} = s_{0} = s. Note that supp(s_{j}) ⊂ D_{j}. Since D_{i} ∩ D_{j} has empty interior for 0 ≤ i < j < m, it follows s_{i}s_{j} = s_{j}s_{i} for any i, j ≥ 1. Thus the s_{i} generate a free abelian group of rank m and that the group ℤ_{m} = ⟨r|r^{m}⟩ acts on this group by cyclically permuting the generators s_{i}. Hence we obtain an action of H_{m} on D. Clearly the elements g = s_{1} ⋯ s_{m} and r commute. The subgroup of H_{n} generated by g, r is isomorphic to ℤ ⊕ ℤ_{m} and its action on D is the same as that constructed in 2.1. It follows that the H_{m} action is chaotic since the periodic points for the H_{m} action is the same as for the action of ℤ⊕ℤ_{m} ≅ ⟨g, r⟩ ⊂ H_{m} action. Since the boundary of D is stable under the action of H_{m}, we obtain an action of H_{m} on D/∂D ≡ S^{2}. Clearly this action is chaotic on S^{2}. (ii). Using the notation introduced in the proof of 2.1(ii), let e_{i,j} be the homeomorhphism of the 2-torus T defined by e_{i,j}(z) = r^{i}s^{j}fs^{−j}r^{−i}(z), for z ∈ D_{i,j}, e_{i,j}(z) = z, for z ∉ D_{i,j}, 0 ≤ i < m, 0 ≤ j < n. Then the e_{i,j} generate a free abelian group of rank mn. Conjugation by the homeomrphisms r and s permute the e_{i,j} in such a manner that the group of homeomorphisms of T generated by the e_{i,j}, 0 ≤ i < m, 0 ≤ j < n, r, s is the semidirect product ℤ^{mn} ⋊ (ℤ_{m} ⊕ ℤ_{n}) = H_{m,n}. The proof that H_{m,n} acts chaotically is exactly similar to part (i) above. □ Consider the usual SL_{n}(ℤ) action on the torus 𝕋 = ℝ^{n}/ℤ^{n}. As observed by Cairns et al. [2], this action is chaotic. Our proof of theorem 2.3 involves a well-known property of a set of reals which are ℚ-linearly independent. Notation: The group SL_{n}(ℤ) acts on the left of ℝ^{n} and hence on the left of the n-torus ℝ^{n}/ℤ^{n} = 𝕋. The element of 𝕋 determined by an x = (x_{1}, ⋯, x_{n}) ∈ ℝ^{n} will be written as [x] = [x_{1}, ⋯, x_{n}]. For g ∈ SL_{n}(ℤ), x ∈ ℝ^{n}, the element g(x) is the product of g with the column vector x^{t}.

Proof of Theorem 2.3: Let G be any finite index subgroup of SL_{n}(ℤ). Consider the G action on the kn-dimensional torus X = 𝕋^{k}, 𝕋 = ℝ^{n}/ℤ^{n}, obtained from restricting the diagonal action of SL_{n}(ℤ) on X. We claim that this action of G on X is chaotic. Indeed we need only verify topological transitivity. Replacing G by a (finite index) subgroup of G if necessary, we may as well assume that G is normal in SL_{n}(ℤ). For 1 ≤ i, j ≤ n, i ≠ j let E_{i,j} denote the n × n matrix whose enrty in k-th row, l-th column is δ_{i,k}δ_{j,l}. Since the element 1+E_{1,2} generates an infinite cyclic group, and since G is of finite index in SL_{n}(ℤ), there exists a d ≥ 2 such that 1 + dE_{1,2} ∈ G. Since G is normal, conjugating 1 + dE_{1,2} by suitable (signed) permutation matrices, it is seen that 1 + dE_{i,j} ∈ G for all i ≠ j, 1 ≤ i, j ≤ n. Let x = [x_{1}, ⋯, x_{k}] ∈ X, x_{i} = [x_{i,1}, ⋯, x_{i,n}] ∈ 𝕋, where the elements x_{i,j} ∈ ℝ, 1 ≤ i ≤ k, 1 ≤ j ≤ n and 1 form a ℚ-linearly independent subset of the reals. Claim: The G orbit of x is dense in X. To prove the claim, let U = ∏_{1≤i≤k} U_{i}, where U_{i} = ∏_{1≤j≤n} U_{i,j} ⊂ T, U_{i,j} being any basic open set in ℝ/ℤ. We need to find an element g ∈ G such that g(x) ∈ U. Since 1, x_{1,2}, ⋯, x_{k,2} are linearly independent over ℚ, the set {[rx_{1,2}, ⋯, rx_{k,2}] | r ∈ dℤ} is a dense subset of ℝ^{k}/ℤ^{k}. It follows that translation of this set by [x_{1,1}, ⋯, x_{1,k}] is also dense in ℝ^{k}/ℤ^{k}. Therefore we can find an r_{1} ∈ ℤ such that y_{i,1} := x_{i,1} + r_{1}dx_{i,2} ∈ U_{i,1} for 1 ≤ i ≤ k. Having obtained r_{j−1} ∈ ℤ such that y_{i,j−1} = x_{i,j−1}+r_{j−1}dx_{i,j} ∈ U_{i,j−1}, 1 ≤ i ≤ k, 2 ≤ j < n, inductively, we use ℚ -linear independence of 1, x_{1,j}, ⋯, x_{k,j}, to obtain an r_{j} ∈ ℤ such that y_{i,j} := x_{i,j} + r_{j}dx_{i,j+1} ∈ U_{i,j}, 1 ≤ i ≤ k. Finally, since 1, y_{1,1}, ⋯, y_{k,1} are ℚ-linearly independent, we can find an r_{n} ∈ ℤ such that y_{i,n} := r_{n}dy_{i,1} + x_{i,n} ∈ U_{i,n}, 1 ≤ i ≤ k. Now the element y = (y_{1}, ⋯, y_{k}) ∈ U ⊂ 𝕋^{k} = X, where y_{i} = [y_{i,1}, ⋯, y_{i,n}] ∈ 𝕋. Write e_{i,j} = 1 + dE_{i,j} ∈ G. Then, for 1 ≤ i ≤ k, e_{n,1}^{r_{n}}e_{n−1,n}^{r_{n−1}} ⋯ e_{1,2}^{r_{1}}(x_{i}) = e_{n,1}^{r_{n}}e_{n−1,n}^{r_{n−1}} ⋯ e_{2,3}^{r_{2}}([y_{i,1}, x_{i,2}, ⋯, x_{i,n}]) = ⋮ = e_{n,1}^{r_{n}}([y_{i,1}, ⋯, y_{i,n−1}, x_{n}]) = [y_{i,1}, ⋯, y_{i,n}]. Hence writing g = e_{n,1}^{r_{n}}e_{n−1,n}^{r_{n−1}} ⋯ e_{1,2}^{r_{1}}, we obtain g(x) = y ∈ U, completing the proof. □ Remark: It follows from Theorem 2.3 that any free group of finite rank acts chaotically on the 2k dimensional torus as such a group can be imbedded as a finite index subgroup of SL_{2}(ℤ). A result of Cairns and Kolganova says that any countably generated (nontrivial) free group can be made to act chaotically on any compact connected triangulable manifold of dimension at least two. (See remark (i) following theorem 2.5 below.) Let k ≥ 2. Denote by X^{(k)} the k-th symmetric power of X; thus it is the quotient of X^{k} by the group permutation group P erm(k) which on X^{k} acts by permuting the coordinates. Note that if X is a 2-manifold then X^{(k)} is a 2k-dimensional manifold. If G acts on X, then the diagonal action of G on X^{k} induces an action on X^{(k)}.

Theorem 2.5 There exists a chaotic ℤ action on the complex projective space ℂP^{k} for any k ≥ 1.

Proof: One has the chaotic ℤ action on ℂP^{1} ≅ S^{2} =: X constructed by Cairns et al.. As observed in the proof of 2.1, this action is topologically mixing and hence, by applying lemma 2.4(ii), the diagonal action of ℤ on X^{k} is chaotic. The induced ℤ-action on the symmetric product ℂP^{k} ≅ (S^{2})^{(k)} is easily seen to be chaotic. □

Remark: Cairns and Kolganova [3] have constructed ℤ actions, indeed, any countably generated free group actions, on any compact connected triangulable manifold M of dimension d ≥ 2. This is achieved by first obtaining such an action on the d-dimensional disk on whose boundary the group acts as identity. The required action on M is obtained via semiconjugacy as M can be realized as the space obtained from the d-dimensional disk by suitable boundary identifications. (See proof of our theorem 2.1 for chaotic ℤ action on even dimensional disks.) However, it would be interesting to have a more direct, “explicit,” description of chaotic group actions on specific manifolds. Note added on 17th May 1999: Theorem 2.1 can be improved to show Theorem:Any finitely generated infinite abelian group can be made to act chaotically on an even dimensional sphere. Proof: We use the notation of theorem 2.1 and its proof. Let G = ℤ^{l} ⊕ F. We assume 1 ≤ l < k. Let G̃ = ℤ^{k} ⊕F. Consider the chaotic G̃-action on the k-fold product ∏_{1≤i≤k} D_{i}, where each D_{i} is a copy of the 2-disk D, obtained as the product of the chaotic ℤ⊕ℤ_{n_{i}} actions on the D_{i}, 1 ≤ i ≤ k, constructed in the proof of theorem 2.1. One has the induced action of the diagonal copy ℤ ≅ Δ ⊂ ℤ^{k} ⊕ 0 ⊂ G̃ on ∏_{1≤i≤k} D_{i,0} ⊂ ∏_{1≤i≤k} D_{i}, where D_{i,0} ⊂ D_{i} is the 2π/n_{i}-sector used in the construction of the chaotic ℤ ⊕ ℤ_{n_{i}} action on D_{i}. Using the fact that the ℤ-action on the 2-disk D constructed by Cairns et al [2] is topologically mixing and applying lemma 2.4(ii) we see that the Δ action on ∏ D_{i,0} is chaotic. Since the traslates of ∏ D_{i,0} by elements of F cover ∏ D_{i}, it follows that the action of Δ⊕F got by restricting the G̃ action on ∏ D_{i} is chaotic. Imbed G in G̃ such that Δ⊕F ⊂ G ⊂ G̃. The resulting G action on ∏ D_{i} is then chaotic. The required chaotic G action on the sphere S^{2k} ≅ ∏ D_{i}/∂(∏ D_{i}) is obtained by semiconjugacy. □

3 Chaos on open manifolds

Let X be a non compact, locally compact, locally connected Hausdorff space. For a homeomorphism f of X, the support supp(f) of f is defined to be the set {x ∈ X | f(x) ≠ x}. Denote by G_{0}(X), the subgroup of those homeomorphisms f of X which are compactly supported, i.e., the closure of supp(f) is a compact subset of X. In this section, we obtain some general results concerning chaotic group actions on X. The most interesting example of such a space is an open manifold. Their importance justifies our choice of the title for this section. We begin with some elementary observations.

Lemma 3.1 Let G be any group that acts effectively on a topological space X. Then

(i) supp(fgf^{−1}) = f(supp(g)), for any f, g ∈ G, (ii) supp(fg) ⊂ supp(f) ∪ supp(g) for any f, g ∈ G. (iii) If S ⊂ G, then supp(S) = supp(H), where H is the subgroup generated by S. Here supp(S) denotes the set ⋃_{g∈S} supp(g). (iv) If H is normal in G, then supp(H) is a G-invariant open subset of H.

Proof: Statements (i) and (ii) are trivial to prove. (iii) follows from (ii) and the obvious fact that supp(f) = supp(f^{−1}). (iv) follows from (i). □

Theorem 3.2 Let X be a locally connected, locally compact, non-compact, Hausdorff topological space and let G a subgroup of G_{0}(X). Suppose that G acts transitively on X. Then (i) If the set of periodic points is dense in X, then X is connected. (ii) supp(S) = ⋃_{g∈S} supp g is dense in X for any set S ⊂ G that generates G. In particular the group G is not finitely generated. (iii) Nontrivial normal subgroups of G are not finitely generated. (iv) The centre of G is trivial. In particular, G is not nilpotent.

Proof: (i) Since X is locally connected, its connected components are open in X. Suppose that X is not connected. Using topological transitivity of the G action, it is easy to see that given any two distinct connected components U and V of X, there exists a g ∈ G such that g(U) = V. Since g is compactly supported, it follows that U ⊂ supp(g) is relatively compact. Since U is also closed, it follows that U must be compact. Thus every component of X is compact. Since X itself is non compact, X must have infinitely many connected components. On the other hand, since G acts transitively on the set of connected components of X, it follows that no element of X can have finite orbit. This contradicts our hypothesis that periodic points for the G action is dense in X. Hence X must be connected. (ii) By lemma 3.1(iii) and (iv), we see that supp(S) is a G-invariant open subset of X. Since the action of G is chaotic, supp(S) must be dense. Now, if S were finite, then X would be a finite union of compact subsets of X, namely the closure of the supports of elements of S, contradicting the assumption that X is not compact. Hence S has to be finite. (iii) Proof is similar to (ii), using lemma 3.1(iv). (iv) To show that the centre of G is trivial, let, if possible, g ∈ G, g ≠ 1 be in the centre of G. Since g ≠ 1, there must an open set U ⊂ supp(g) such that U ∩ g(U) = ∅. Let V be an open set disjoint from supp(g). By topological transitivity, we can find h ∈ G such that hU ∩V ≠ ∅. Let y ∈ hU ∩V so that u := h^{−1}(y) ∈ U. Now, g = hgh^{−1} implies y = g(y) = hgh^{−1}(y) = hg(u). Hence h^{−1}(y) = g(u) ∈ g(U). This is a contradiction since U ∩ g(U) = ∅. □ As an immediate consequence abelian subgroups of G_{0}(X) cannot act chaotically on X. We shall later see that if a subgroup G of G_{0}(X) acts chaotically on X, then G cannot even have a finite index solvable subgroup. In other words, virtually solvable subgroups of G_{0}(X) cannot act chaotically on X. Let G be group acting chaotically and effectively on a Hausdorff space Y. Let H be a subgroup of G. Then H, being residually finite, acts chaotically and faithfully on some Hausdorff space. It is however not true that the action of H obtained by restriction of the G-action on X is chaotic. For example there exists a chaotic action of the group ℤ × ℤ_{2} × ℤ_{2} on the 2-torus such that no cyclic subgroup acts chaotically on the torus (see [2], cf. Theorem 2.1(ii)). However, in our case we can prove the following.

Theorem 3.3 Let X be locally compact, non compact, Hausdorff space. Let G act chaotically on X and let H be a subgroup of G such that for some set S of distinct left coset representatives of H in G, the set supp(S) = ⋃_{s∈S} supp(s) is not dense in X. Then the H action on X by restriction is chaotic.

Proof: We need only check topological transitivity for the H action on X. Let U be any non empty open set in X which does not meet A := supp(S). Set W_{s} = ⋃_{h∈H} sh(U), s ∈ S. (We shall assume that 1 ∈ S.) We claim that W_{1} − A = W_{s} − A for any s ∈ S. To see this note that s(W_{1}) = W_{s} and s(A) = A for all s ∈ S. It follows that W_{1} − A = W_{s} − A for all s ∈ S. We claim that W_{1} is dense in X. Note that this would imply topological transitivity of the H-action. To prove the claim, observe that W := ⋃_{s∈S} W_{s} is dense in X since G acts chaotically on X. In particular, W − A = ⋃_{s∈S}(W_{s} − A) = W_{1} − A is dense in X − A. Therefore we need only show, for any open set V ⊂ A, that W_{1} ∩ V is non-empty. Let g ∈ G be such that V ∩ g(U) ≠ ∅. Writing g^{−1} = sh with s ∈ S and h ∈ H, we see that V ∩ hs(U) ≠ ∅. Since U ∩ A = ∅ and supp(s) ⊂ A, we have s(U) = U so that V ∩ h(U) ≠ ∅. It follows that V ∩ W_{1} ≠ ∅ as h(U) ⊂ W_{1}. □

Corollory 3.4 Let G ⊂ G_{0}(X) act chaotically on X and let H be a finite index subgroup of G. Then H acts chaotically on X. If H′ is any subgroup of G which acts chaotically on X, then the centralizer of H′ in G is trivial.

Proof: Let H be a finite index subgroup of G and let S be a (finite) set of distict left coset rerpresentatives for H in G. Then A = ⋃_{s∈S} supp(s) is relatively compact as S is finite and each g ∈ G is compactly supported. Since X is non-compact, it follows that A is not dense in X. Hence H acts chaotically. Let g ∈ G, g ≠ 1 commute with every element in H′. Then the subgroup K =< H′, g >⊂ G acts chaotically on M. For, topological tranisitivity of H′ implies the topological transitivity of K, the densitity of periodic points for the K action follows from the corresponding property for the G action. As g ∈ K is in the center of K this contradicts Theorem 3.2. □

Proposition 3.5 Let H and H′ be normal subgroups of G ⊂ G_{0}(X). Suppose G, H and H′ act chaotically on X. Then the group [H, H′] ⊂ G acts chaotically on X. In particular, no finite index subgroup of G can be solvable.

Proof: We need only show that the K := [H, H′]-action is topologically transitive. Let U and V be any two disjoint open sets in X. Let h ∈ H be such that h(U) ∩ V is non-empty. Let U′ = h^{−1}(V) ∩ U. Let W be any open set outside the support of h ∈ H. Let h′ ∈ H′ be such that U′ ∩ h′(W) is non-empty. Let U′′ = U′ ∩ h′(W), so that h^{′−1}(U′′) ⊂ W. Now hh′h^{−1}h^{′−1}(U′′) = h(U′′) ⊂ V since h(w) = w for all w ∈ W. Hence [h, h′](U) ∩ V is non-empty. If K is a finite index subgroup of G, then K action (via restriction) on X is chaotic. It follows that K′ = [K, K] ⊂ K again acts chaotically on X. Repeated application of the same argument shows that K cannot be solvable. □

Proposition 3.6 Let G ⊂ G_{0}(X) act chaotically on X. Suppose that G decomposes as a nontrivial product H × H′. Then neither H nor H′ acts topologically transitively on X.

Proof: Let, if possible, the H-action be topologically transitive. Then so is the action of K = ⟨H, h′⟩ for any 1 ≠ h′ ∈ H′. Since K ⊂ G, the periodic points for the K action is dense in M. It follows that K acts chaotically on M. Clearly the h′ is in the center of K which contradicts 3.2. □ Definition: We shall say that the group G satisfies property C if there exists a countable collection {N_{i}}_{i≥1} of finite index normal subgroups N_{i}, such that ∩_{i}N_{i} = {1}. Clearly G is residually finite if has property C. We state without proof the following theorem, which implies that any group that acts chaotically on a connected open manifold must have property C. Proof that (ii) implies (i) uses the observation that if G acts chaotically on a second countable Hausdorff space X, then there exists a countable family of finite orbits whose union is dense in X.

Theorem 3.7 For a group G the following are equivalent. (i) G satisfies C. (ii) G acts chaotically and faithfully on a second countable Hausdorff X. □

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