Foliations with measure preserving holonomy

Plante, J. F.

doi:10.1007/bfb0082576

Cited by 145 publications

(97 citation statements)

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“…Haefliger shows in [7, II.6] that TT^BYn) = 0 for 0 ^ i < n. One can easily see that 7r n does not vanish by noting that there is a characteristic class fx e H n (BTa ; R) which is not induced from H n (BSIj(n, R)). The value of n for the pair (^, co) is the element oiH n (M ; R) represented by the closed form u> on M. (For some applications of a generalization of this characteristic class see [13].) Using Haefliger's methods together with Gromov's theorem [5] about the extension of divergence-free vector fields, we will prove LEMMA 1.…”

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confidence: 97%

On Groups of Volume-Preserving Diffeomorphisms and Foliations with Transverse Volume Form

McDuff

1981

Proceedings of the London Mathematical Society

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Let Q be a smooth nowhere-zero w-form on a non-compact ndimensional manifold Y. We study the homology of the group Diff n Y, consisting of all diffeomorphisms of Y preserving Q, provided with the discrete topology. In particular, if Q is the standard volume form dx { A ... A dx n on R", and if n # 2,3, we show that the natural maps #Diff n R" c > £Emb n R" -^-> BTJ induce isomorphisms of integral homology. Here Emb n R" is the discrete monoid of smooth embeddings of R" preserving 0., and J5F n " is the Haefliger classifying space for codimension-n smooth foliations with a smooth non-vanishing transverse closed w-form. (Analogous statements for the group of all diffeomorphisms of R" were proved by Segal in [16] with no restriction on dimension.) It follows that if n # 2, 3 the natural map j5Diff n R" -> l?SO(w), where SO(w) is given its usual topology, induces an isomorphism on integral homology #; for 0 ^ i < n, and a surjection on H n with kernel isomorphic to R. If Y is a non-compact manifold which is diffeomorphic to the interior of a compact n-dimensional manifold with boundary, and if every end of Y has infinite Q-volume, we show that providing that n # 2,3. Here Diff fi0 <= Diff n denotes the subgroup consisting of elements smoothly isotopic to the identity.

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On Groups of Volume-Preserving Diffeomorphisms and Foliations with Transverse Volume Form

McDuff

1981

Proceedings of the London Mathematical Society

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“…The following measure theoretic argument is similar to that used by Plante in proving Proposition 5.3 of [8]. The following measure theoretic argument is similar to that used by Plante in proving Proposition 5.3 of [8].…”

Section: Theorem 2 Ifg Is An Amenable Group Then G Is Cyclically Rimentioning

confidence: 87%

Amenability and Right Orderable Groups

Kropholler

1993

Bulletin of the London Mathematical Society

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“…Following Plante [13], this implies the exponential growth for every leaf of a P-solenoid. The existence of a foliated cycle is a very strong hypothesis.…”

Section: On Invariant Measures Of Finite Affine Type Tilings 1165mentioning

confidence: 99%

On invariant measures of finite affine type tilings

Petite¹

2006

Ergod. Th. Dynam. Sys.

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In this paper, we consider tilings of the hyperbolic 2-space H 2 , built with a finite number of polygonal tiles, up to affine transformation. To such a tiling T , we associate a space of tilings: the continuous hull (T ) on which the affine group acts. This space (T ) inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First, we prove that the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally, we give examples with an arbitrary number of ergodic invariant probability measures.

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Foliations with measure preserving holonomy

Cited by 145 publications

References 1 publication

On Groups of Volume-Preserving Diffeomorphisms and Foliations with Transverse Volume Form

On Groups of Volume-Preserving Diffeomorphisms and Foliations with Transverse Volume Form

Amenability and Right Orderable Groups

On invariant measures of finite affine type tilings

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