Finite dimensional representations and subgroup actions on homogeneous spaces

LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/G _ℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/G _ℤ; that ifH is epimorphic inG then the action ofH onG/G _ℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/G _ℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.     

Affine embeddings with a finite number of orbits

LetG be a reductive algebraic group and letH be a reductive subgroup ofG. We describe all pairs (G, H) such that, for any affineG-varietyX with a denseG-orbit isomorphic toG/H, the number ofG-orbits inX is finite.Work of both authors was supported by INTAS-OPEN-97-1570, by CRDF grant RM1-2088, and by RFBR grant 01-01-00756.     

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

Growth of homogenous spaces,density of discrete subgroups and Kazhdan's property (T)

Summary LetG be a semisimple Lie group with finite center and no compact factors. We show that ifH is a closed unimodular subgroup ofG such thatG/H has subexponential volume growth, thenH is Zariski dense inG. Moreover, ifG has Kazhdan's property (T) thenG/H must have finite volume. We extend these results to semisimple groups over a local field.Oblatum 5-VII-1991 & 2-I-1992This work was supported by an NSF Postdoctoral Research Fellowship     

Remarks on the wonderful compactification of semisimple algebraic groups

We prove that ifG is a semisimple algebraic group of adjoint type over the field of complex numbers,H is the subgroup of all fixed points of an involution σ ofG that is induced by an involution σ of the simply connected coveringĜ ofG, then the wonderful compactification of the homogeneous spaceG/H is isomorphic to the G.I.T quotientG ^ss (L)//H of the wonderful compactificationG ofG for a suitable choice of a line bundleL onG. We also prove a functorial property of the wonderful compactifications of semisimple algebraic groups of adjoint type.     

Some remarks on automorphisms of bounded displacement and bounded cocycles

Some time agoTits has considered the question of characterizingB(G), and more generally, of finding the automorphisms of bounded displacement ofG, particularly whenG is a connected real Lie group. Our purpose here is to extend these results to various other cases as well as to deal with the analogous questions for 1-cocycles. We concern ourselves, among other things, with the question of sufficient conditions forB(G)=Z(G), the center, or more generally, forG to have no non-trivial automorphisms of bounded displacement. The significance of such conditions can be seen in work of the author together withF. Greenleaf andL. Rothschild where, for example, the Selberg form of the Borel density theorem is considerably generalized. These conditions are therefore closely related to, but not identical with, sufficient conditions for the Zariski density of a closed subgroupH ofG withG/H having finite volume, see [11]. For this reason it is enlightening to compare these results with those of [11]. On the cocycle level, we give sufficient conditions for the points with bounded orbit under a linear representation to be fixed and more generally, for a bounded 1-cocycle to be identically zero. These conditions actually play a role in [11]; they are among the sufficient conditions necessary to establish Zariski density ofH inG. We also deal with certain converse questions and applications to homogeneous spaces of finite volume. For example, ifG/H has finite volume and is an automorphism ofG leavingH pointwise fixed, then has bounded displacement. If is a 1-cocycle and /H is trivial, then is itself bounded.Research partially supported by NSF Grant=MPS 75-08268.     

On the number of copies of one hypergraph in another

Given two hypergraphsH andG, letN(G, H) denote the number of subhypergraphs ofG isomorphic toH. LetN(l, H) denote the maximum ofN(G, H), taken over allG with exactlyl edges. In [1] Noga Alon analyzes the asymptotic behaviour ofN(l, H) forH a graph. We generalize this to hypergraphs: Theorem:For a hypergraph H with fractional cover number ρ*,N(G,H).=θ(lρ*) The interesting part of this is the upper bound, which is shown to be a simple consequence of an entropy lemma of J. Shearer. In a special case which includes graphs, we also provide a different proof using a hypercontractive estimate. Research supported in part by NSF.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Height in splittings of hyperbolic groups

SupposeH is a hyperbolic subgroup of a hyperbolic groupG. Assume there existsn > 0 such that the intersection ofn essentially distinct conjugates ofH is always finite. Further assumeG splits overH with hyperbolic vertex and edge groups and the two inclusions ofH are quasi-isometric embeddings. ThenH is quasiconvex inG. This answers a question of Swarup and provides a partial converse to the main theorem of [23].     

Topologisch quasinormale Untergruppen zusammenhängender lokalkompakter Gruppen

A closed subgroupQ of a topological groupG is called topologically quasinormal (tqn) inG if holds for every closed subgroupA ofG. We show that every tqn subgroup of a connected locally compact group is actually a normal subgroup. Besides we prove: a homogeneous spaceG/H of a connected Lie groupG with the property that every non-trivial one-parameter orbit is dense has dimension at most one.     

A note on a theorem of W. Gaschütz and N. Itô

Given a finite groupG andp an odd prime number, we conclude thatO ^p(G)G isp-nilpotent when for every subgroupH ofG of orderp there exists a subgroupK ofG such thatG=HK andH permutes with every subgroup ofK.     

Virtually free factors of pro-p groups

Letp be a prime number,G a pro-p group, andH a closed (topologically) finitely generated subgroup ofG. We give conditions under whichH is virtually a free factor ofG, i.e., that there exists an open subgroupU ofG such thatU is the free pro-p product ofH and some other subgroup ofU. We prove that this happens if eitherG is a free pro-p group of any rank, or ifG is a free pro-p product of finitely generated pro-p groups. Research supported in part by grants from NSERC (Canada) and DGICYT (Spain).     

p-Pseudomeasures and closed subgroups

LetPM _p (G) be the space of allp-pseudomeasures on a locally compact groupG. We show the existence of a conditional expectation fromPM _p (G) ontoPM _p (H) whereH is a closed normal subgroup ofG. As an application we give a new proof of the fact thatH is a set ofp-synthesis inG; we also get an inequality involving the operator norm of bounded measures onG. Moreover, in analogy with a theorem of Reiter, we obtain a result concerning the closed ideals of the Figà-Talamanca Herz algebra ofG.     

Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

Continuity of operators commuting with translations for compact groups

In this note we show for certain Frechet spacesF(G) of functions (distributions) on a compact groupG that if every translation invariant linear functional onF(G) is continuous then every linear operatorT:F(G)F(G) commuting with translations is continuous. This solves partially a problem in [7] ofG. H. Meisters and improves the result [5] ofC. J. Lester. An application for compact groups which do not have the mean zero weak containment property follows by the result [10] ofG. A. Willis.     

Semigroup actions on homogeneous spaces

LetG be a connected semi-simple Lie group with finite center andS⊄G a subsemigroup with interior points. LetG/L be a homogeneous space. There is a natural action ofS onG/L. The relationx≤y ify ∈Sx, x, y ∈G/L, is transitive but not reflexive nor symmetric. Roughly, a control set is a subsetD ⊄G/L, inside of which reflexivity and symmetry for ≤ hold. Control sets are studied inG/L whenL is the minimal parabolic subgroup. They are characterized by means of the Weyl chambers inG meeting intS. Thus, for eachw ∈W, the Weyl group ofG, there is a control setD _w .D ₁ is the only invariant control set, and the subsetW(S)={w:D _w =D ₁} turns out to be a subgroup. The control sets are determined byW(S)/W. The following consequences are derived: i)S=G ifS is transitive onG/H, i.e.Sx=G/H for allx ∈G/H. HereH is a non discrete closed subgroup different fromG andG is simple. ii)S is neither left nor right reversible ifS #G iii)S is maximal only if it is the semigroup of compressions of a subset of some minimal flag manifold. Research partially supported by CNPq grant n^o 50.13.73/91-8     

Classes caracteristiques de Γ(G,H)-structures et finitude de leur evaluation

LetG be a Lie group andH a closed subgroup ofG. We denote by (G,H) the groupoïd of germs of left translations ofG over the homogeneous spaceG/H.LetV be a compact manifold andx the universal characteristic class of dimensionk which belongs to the vector spaceH _cont ^k ((G, H)).The evaluation ofx over all the (G, H)-structures overV determines a subsetA _{(G, H)} (x, V) of the vector spaceH ^k (V;).We show that in some cases this set is finite.     

Some conditions which almost characterize Frobenius groups

The main result of this paper is the following: LetG be a group with a proper non-trivial normal subgroupH such that each coset ofH distinct fromH is contained in a conjugacy class ofG. IfG is not a Frobenius group with kernelH then one ofH orG/H is ap-group. The hypothesis of this theorem is shown to be equivalent to a condition on characters ofG. The only group the author knows which satisfies this hypothesis and is not either Frobenius or ap-group is one of order 72.