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1.
We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫
K
kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q.
Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999 相似文献
2.
A. Dranishnikov 《Geometriae Dedicata》2009,141(1):59-86
We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension
asdim
Z
X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology
is strictly less than its asymptotic dimension.
相似文献
3.
Gordan Savin 《Israel Journal of Mathematics》1992,80(1-2):195-205
LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L
2(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP
f(g)=ΣH∩ΓΓ
f(γ{}itg) is an automorphic form on Γ\G. We show thatP
f is cuspidal in some cases, whenH ∩Γ\H is compact.
Partially supported by NSF Grant # DMS 9103608. 相似文献
4.
Let U := L\G be a homogeneous variety defined over a number field K, where G is a connected semisimple K-group and L is a connected maximal semisimple K-subgroup of G with finite index in its normalizer. Assuming that G(K
v
) acts transitively on U(K
v
) for almost all places v of K, we obtain an asymptotic for the number of rational points U(K) with height bounded by T as T → ∞, and settle new cases of Manin’s conjecture for many wonderful varieties. The main ingredient of our approach is the
equidistribution of semisimple adelic periods, which is established using the theory of unipotent flows. 相似文献
5.
C S Rajan 《Proceedings Mathematical Sciences》1994,104(2):389-395
LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL
2(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic
groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston
that this is true for all cocompact lattices inSL(2, C)).
We also show thatG acts trivially on the coherent cohomology groupsH
i(Γ/G, O) for anyi≥0. 相似文献
6.
Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of
a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups.
(Received 13 June 1998; in revised form 31 March 1999) 相似文献
7.
Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. 相似文献
8.
Nir Ben David 《Israel Journal of Mathematics》2009,170(1):317-335
A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class [c] ∈ H
2(G, ℂ*) (G acts trivially on ℂ*). Groups of central type play a fundamental role in the classification of semisimple triangular complex
Hopf algebras and can be determined by their representation-theoretical properties.
Suppose that a finite group Q acts on an abelian group A so that there exists a bijective 1-cocycle π ∈ Z
1(Q,Ǎ), where Ǎ = Hom(A, ℂ*) is endowed with the diagonal Q-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in Z
2(G, ℂ*), where G:= A × Q. Hence, the semidirect product G is of central type.
In this paper, we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular,
given a bijective class [π] ∈ H
1(Q,Ǎ) as above, we construct non-degenerate classes [cπ] ∈ H
2(G,ℂ*) for certain extensions 1 → A → G → Q → 1 which are not necessarily split. We thus strictly extend the above family of
central type groups. 相似文献
9.
Ursula Hamenstädt 《Geometric And Functional Analysis》2009,19(1):170-205
Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H2cb(G, Lp(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X).
Partially supported by Sonderforschungsbereich 611. 相似文献
10.
V. Stakėnas 《Lithuanian Mathematical Journal》2006,46(2):208-216
Let Q
+ denote the set of positive rational numbers. We define discrete probability measures ν
x
on the σ-algebra of subsets of Q
+.We introduce additive functions ƒ: Q
+ → G and obtain a bound for νx(ƒ (r) ∉ X+X−X) using a probability related to some independent random variables. This inequality is an analogue to that proved by I. Ruzsa
for additive arithmetical functions.
Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 256–266, April–June, 2006. 相似文献
11.
Let M = G/K be a homogeneous differentiable manifold. We consider the homogeneous bundle = (G, π, G/K, K) and the tangent bundle τ
G/K of M = G/K, and give some results about the existence of homogeneous vectors on the fiber space of τ
G/K, for both cases of G semisimple and weakly semisimple.
相似文献
12.
For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory. 相似文献
13.
Let G = ℤ
p
, p an odd prime, act freely on a finite-dimensional CW-complex X with mod p cohomology isomorphic to that of a lens space L
2m−1(p; q
1, …, q
m
). In this paper, we determine the mod p cohomology ring of the orbit space X/G, when p
2 ∤ m. 相似文献
14.
Mladen Božičević 《Mediterranean Journal of Mathematics》2007,4(4):407-418
Let G be a real Lie group acting on real analytic manifold with finitely many orbits. We prove that the characteristic cycle map
is a surjective homomorphism from the K-group of G-equivariant sheaves on X to the top homology group of the conormal variety of the G-action on X. We also show that the top homology group of the G-action on X is a free -module of rank equal to the number of G-orbits.
This work was completed with the support of the Ministry of Science, Education and Sport of Croatia, and Infodesign d.o.o.,
Varaždin. 相似文献
15.
L. Barchini 《Mathematische Annalen》2003,326(2):331-346
Let G be a connected semisimple Lie group contained in its simply connected complexification G
C
. Let KG∩K
C
be a maximal compact subgroup of G. Denote by X
o
the unique closed G-orbit in the full flag manifold ℱ and by 𝒪 the unique open K
C
-orbit in ℱ. The set consisting of the elements gK
C
so that gX
o
⊂𝒪 is an Stein extension of G/K⊂G
C
/K
C
. There is a universal domain , natural form the point of view of group actions which has been conjectured to be Stein. The main result of this paper is
the inclusion . In the second part of the paper I show, under some dominance condition in the parameter, that representations in Dolbeault
cohomology can be realized as holomorphic sections of vector bundles over .
Received: 9 September 2002 / Revised version: 12 July 2002 /
Published online: 8 April 2003
Mathematics Subject Classification (2002): 22E30
Research partially supported by NSF grant DMS-9801605 and DMS 0074991. 相似文献
16.
Guido Pezzini 《Transformation Groups》2009,14(3):677-694
Let G be a complex semisimple linear algebraic group, and X a wonderful G-variety. We determine the connected automorphism group Aut0(X) and we calculate Luna’s invariants of X under its action. 相似文献
17.
Harald Grobner 《Monatshefte für Mathematik》2010,139(1):335-340
Let
G/\mathbb Q{G/\mathbb Q} be the simple algebraic group Sp(n, 1) and G = G(N){\Gamma=\Gamma(N)} a principal congruence subgroup of level N ≥ 3. Denote by K a maximal compact subgroup of the real Lie group
G(\mathbb R){G(\mathbb R)} . Then a double quotient
G\G(\mathbb R)/K{\Gamma\backslash G(\mathbb R)/K} is called an arithmetically defined, quaternionic hyperbolic n-manifold. In this paper we give an explicit growth condition for the dimension of cuspidal cohomology
H2ncusp(G\G(\mathbb R)/K,E){H^{2n}_{cusp}(\Gamma\backslash G(\mathbb R)/K,E)} in terms of the underlying arithmetic structure of G and certain values of zeta-functions. These results rely on the work of Arakawa (Automorphic Forms of Several Variables:
Taniguchi Symposium, Katata, 1983, eds. I. Satake and Y. Morita (Birkh?user, Boston), pp. 1–48, 1984). 相似文献
18.
Bangteng Xu 《Journal of Algebraic Combinatorics》2008,27(2):127-141
Sunder and Wildberger (J. Algebr. Comb.
18, 135–151, 2003) introduced the notion of actions of finite hypergroups, and studied maximal irreducible actions and *-actions. One of the
main results of Sunder and Wildberger states that if a finite hypergroup K admits an irreducible action which is both a maximal action and a *-action, then K arises from an association scheme. In this paper we will first show that an irreducible maximal action must be a *-action,
and hence improve Sunder and Wildberger’s result (Theorem 2.9). Another important type of actions is the so-called w-maximal actions. For a w-maximal action π:K→Aff (X), we will prove that π is faithful and |X|≥|K|, and |K| is the best possible lower bound of |X|. We will also discuss the strong connectivity of the digraphs induced by a w-maximal action. 相似文献
19.
We apply the “homotopy coniveau” machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized
to arbitrary X by Edidin–Graham) and an Atiyah–Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch–Lichtenbaum and Friedlander–Suslin.
The first-named author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and
support of the NSF via grants DMS-0140445 and DMS-0457195. 相似文献
20.
Jan van Mill 《Monatshefte für Mathematik》2009,165(1):257-266
We show that for any sufficiently homogeneous metrizable compactum X there is a Polish group G acting continuously on the space of rational numbers
\mathbbQ{\mathbb{Q}} such that X is its unique G-compactification. This allows us to answer Problem 995 in the ‘Open Problems in Topology II’ book in the negative: there
is a one-dimensional Polish group G acting transitively on
\mathbbQ{\mathbb{Q}} for which the Hilbert cube is its unique G-completion. 相似文献