On the Cohomology of Discrete Abelian Group Actions

We prove that if G is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of G on Π g∈G (X 0 , µ 0) g , for (X 0 , µ 0) an arbitrary probability space, has first cohomology group isomorphic to the character group of G.

Annales Henri Poincaré

We study homeomorphisms of minimal and uniquely ergodic tiling spaces with finite local complexity (FLC), of which suspensions of (minimal and uniquely ergodic) d-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms, and show that all homeomorphisms are a combination of tiling deformations, maps homotopic to the identity (known as quasi-translations), and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology, and show that all orbit equivalences are combinations of tiling deformations, quasi-translations, and topological conjugacies. These generalize results of Parry and Sullivan to higher dimensions. We also show that homeomorphisms (FLC) or orbit equivalences (ILC) are completely parametrized by the appropriate cohomological invariants. Finally, we show that, under suitable cohomological conditions, continuous maps between tiling spaces are homotopic to compositions of tiling deformations and either local derivations (FLC) or factor maps (ILC).

Israel Journal of Mathematics, 2003

Modules of harmonic cochains on the Bruhat Tits building of the projective general linear group over a p-adic field were defined by one of the authors, and were shown to represent the cohomology of Drinfel'd's p-adic symmetric domain. Here we define certain non-trivial natural extensions of these modules and study their properties. In particular, for a quotient of Drinfel'd's space by a discrete cocompact group, we are able to define maps between consecutive graded pieces of its de Rham cohomology, which we show to be (essentially) isomorphisms. We believe that these maps are graded versions of the Hyodo-Kato monodromy operator N.

H n (G, M) where n = 0, 1, 2, 3,. . ., called the nth homology and cohomology of G with coefficients in M. To understand this we need to know what a representation of G is. It is the same thing as ZG-module, but for this we need to know what the group ring ZG is, so some preparation is required. The homology and cohomology groups may be defined topologically and also algebraically. We will not do much with the topological definition, but to say something about it consider the following result: THEOREM (Hurewicz 1936). Let X be a path-connected space with π n X = 0 for all n ≥ 2 (such X is called 'aspherical'). Then X is determined up to homotopy by π 1 (x). If G = π 1 (X) for some aspherical space X we call X an Eilenberg-MacLane space K(G, 1), or (if the group is discrete) the classifying space BG. (It classifies principal G-bundles, whatever they are.) If an aspherical space X is locally path connected the universal cover˜X is contractible and X = ˜ X/G. Also H n (X) and H n (X) depend only on π 1 (X). If G = π 1 (X) we may thus define H n (G, Z) = H n (X) and H n (G, Z) = H n (X) and because X is determined up to homotopy equivalence the definition does not depend on X. As an example we could take X to be d loops joined together at a point. Then π 1 (X) = F d is free on d generators and π n (X) = 0 for n ≥ 2. Thus according to the above definition H n (F d , Z) = Z if n = 0 Z d if n = 1 0 otherwise. Also, the universal cover of X is the tree on which F d acts freely, and it is contractible. The theorem of Hurewicz tells us what the group cohomology is if there happens to be an aspherical space with the right fundamental group, but it does not say that there always is such a space.

Ergodic Theory and Dynamical Systems, 2011

Exact regularity was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider the analog of exact regularity for arbitrary tiling spaces. Let T be a d-dimensional repetitive tiling, and let Ω be its hull. If Ȟd(Ω,ℚ)=ℚk, then there exist k patches each of whose appearances governs the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling T comes from a substitution, then we can quantify that convergence rate. If T is also one dimensional, we put constraints on the measure of any cylinder set in Ω.

Groups, Geometry, and Dynamics, 2000

For finitely generated groups , the isomorphism between the first`p-cohomology H 1 .p/ ./ and the reduced 1-cohomology with coefficients in`p./ is exploited to obtain vanishing results for H 1 .p/ ./. The following cases are treated: groups acting on trees, groups with infinite center, wreath products, and lattices in product groups.

Journal d'Analyse Mathématique, 1994

Let F be a non-elementary finitely generated Kleinian group with region of discontinuity [2. Let q be an integer, q > 2. The group I" acts on the right on the vector space lI2q_ 2 of polynomials of degree less than or equal to 2q-2 via Eichler action. We note by Aq(fl, F) the space of cusp forms for P of weight (-2q) and the space of parabolic cobomology classes by PH I (F, II2q_2). Bets introduced an anti-linear map [3~ : Aq(f~, 1") .... PH I (1", II2q_2). We try to determine the class of Kleinian groups for which the Bers map is surjective. We show that the Bers map is surjective for geometrically finite function groups. We also obtain a characterization of geometrically finite function groups. As an application, we reprove theorems of Maskit on inequalities involving the dimension of the space of cusp forms supported on an invariant component and the dimension of the space of cusp forms supported on the other components for finitely generated function groups. We also show all these inequalities are equalities for geometrically finite B-groups. Preliminaries and summary of results Let 1" be a non-elementary finitely generated Kleinian group with region of discontinuity f~ = f~(I') and limit set A = A(F). Let q be an integer, q > 2. The group 1" acts on the right on the vector space of polynomials of degree less than or equal to 2q-2 via Eichler action: (p."/)(Z)-'p('y(Z))~t(Z) l-q forp E II2q-2 and 7 E F. A mapping X : P ~ II2q-2 is a cocycle if X('Yl o "~2) = X('n) 9 "Y2 + X('r2) for any "yl ,'y2 E F. Such a cocycle is coboundary if X('~) = P-'~-P, "y E i ~ for some fixed p E 172q-2.

Journal of Mathematics Research, 2015

In this paper we obtain a characterization of k-type transitivity for a Z d-action on certain spaces and then prove that k-type SDIC is redundant in the definition of k-type Devaney chaos for Z d-actions on infinite metric spaces. We define different types of chaos for Z d-actions and prove results related to their preservations under conjugacy and uniform conjugacy. Finally we discuss k-type properties on product spaces.

Communications in Mathematical Physics, 2006

The continuous Hull of a repetitive tiling T in R d with the Finite Pattern Condition (FPC) inherits a minimal R d -lamination structure with flat leaves and a transversal T which is a Cantor set. This class of tiling includes the Penrose & the Amman Benkker ones in 2D, as well as the icosahedral tilings in 3D. We show that the continuous Hull, with its canonical R d -action, can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact d-manifolds. As a consequence, the longitudinal cohomology and the K-theory of the corresponding C * -algebra A T are obtained as direct limits of cohomology and K-theory of ordinary manifolds. Moreover, the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of R d . At last, the gap labeling theorem holds: given an invariant ergodic probability measure µ on the Hull the corresponding Integrated Density of States (IDS) of any selfadjoint operators affiliated to A T takes on values on spectral gaps in the Z-module generated by the occurrence probabilities of finite patches in the tiling. ← H d (B n , R), f * n . Then H d ( , R) has a canonical positive cone induced by the orientation of the B n 's, which is in one-to-one correspondence with the space of R d -invariant positive finite measures on . 3. If the f * n 's are uniformly bounded in n, the Hull is uniquely ergodic. If, in addition, dim H d (B n , R) = N < ∞ there is no more than N invariant ergodic probability measures on . 4. Let A = C( ) R d be the crossed product C * -algebra associated with the Hull. Then through the Thom-Connes isomorphism, K * (A) = lim → K * +d (B n ), f * n . 5. Any R d -invariant ergodic probability measure µ defines canonically a trace T µ on A together with an induced measure µ tr on the transversal [16]. Then the image by T µ of the group K 0 (A) coincides with dµ tr C( , Z), namely the Z-module generated by the occurrence numbers (w.r.t. µ) of patches of finite size of L (gap labeling theorem).

Ergodic Theory and Dynamical Systems, 2007

We relate Kellendonk and Putnam's pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives an version of PE cohomology with integer coefficients, or with values in any Abelian group. It also provides an easy proof of Kellendonk and Putnam's original theorem relating PE cohomology to thě Cech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.