Minimality and unique ergodicity of homogeneous actions

Consider a connected Lie groupG, a lattice Γ inG, a connected subgroupH ofG, and the adjoint representation Ad ofG on its Lie algebra g. Suppose that Ad(H) splits into a semidirect product of a reductive subgroup and the unipotent radical. We prove that the minimality of the leftH-action onG/Γ then implies its unique ergodicity. Simultaneously, we suggest a reduction of the study of finite ergodic measures for an arbitrary action (G/Γ,H), where the subgroupH∈G is connected and Γ∈G is discrete, to the case of an Abelian subgroupH. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 293–301, August, 1999.     

A remark on equivariance of a moment mapping

It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension $ \tilde G It is known that the moment mapping of a strongly symplectic action of a Lie group on a symplectic manifold can be non-equivariant. It is proved in the paper that such non-equivariance can be eliminated in a canonical way; namely, a strongly symplectic action G × M → M of a connected Lie group has a Hamiltonian extension × M → M. Original Russian Text ? I.V. Mikityuk, A.M. Stepin, 2008, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2008, Vol. 63, No. 3, pp. 30–33.     

Infinite primitive directed graphs

A group G of permutations of a set Ω is primitive if it acts transitively on Ω, and the only G-invariant equivalence relations on Ω are the trivial and universal relations. A digraph Γ is primitive if its automorphism group acts primitively on its vertex set, and is infinite if its vertex set is infinite. It has connectivity one if it is connected and there exists a vertex α of Γ, such that the induced digraph Γ∖{α} is not connected. If Γ has connectivity one, a lobe of Γ is a connected subgraph that is maximal subject to the condition that it does not have connectivity one. Primitive graphs (and thus digraphs) with connectivity one are necessarily infinite.     

Deformations of complex structures on Γ/SL 2(C)

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.     

Proper actions of lattices on contractible manifolds

Every lattice Γ in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if Γ acts on a contractible manifold W and if either?1) the action is properly discontinuous, or?2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1,?then dimW≥dimG/K. Oblatum 19-I-2001 & 24-IV-2002?Published online: 5 September 2002 RID="*" ID="*"The authors gratefully acknowledge support from the National Science Foundation.     

Up–down representations and ergodic theory in nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G.     

Asymptotically cylindrical 7-manifolds of holonomy <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> with applications to compact irreducible <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>-manifolds

We construct examples of exponentially asymptotically cylindrical (EAC) Riemannian 7-manifolds with holonomy group equal to G ₂. To our knowledge, these are the first such examples. We also obtain EAC coassociative calibrated submanifolds. Finally, we apply our results to show that one of the compact G ₂-manifolds constructed by Joyce by desingularisation of a flat orbifold T ⁷/Γ can be deformed to give one of the compact G ₂-manifolds obtainable as a generalized connected sum of two EAC SU(3)-manifolds via the method of Kovalev (J Reine Angew Math 565:125–160, 2003).     

On the outer automorphism group of a hyperbolic group

LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite. Dedicated to Professor Takushiro Ochiai for his sixtieth birthday     

On graphs related to conjugacy classes of groups

LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at the Tel Aviv University under the supervision of Prof. Marcel Herzog.     

G 2-manifolds and coassociative torus fibration

Let (φ ₀, g ₀) be a flat G ₂-structure on the torus T ⁷. For a certain finite group Γ-action on T ⁷ preserving the G ₂-structure, Joyce constructed a closed G ₂-manifold M from the resolution of the orbifold T ⁷/Γ. The main purpose of this paper is to prove that there exist global coassociative fibrations on open submanifolds of certain Joyce manifolds.      

A Family of Non-quasiprimitive Graphs Constructed From PSU3（q） Where q Is Odd

Let Г be a simple connected graph and let G be a group of automorphisms of Г. Г is said to be （G, 2）-arc transitive if G is transitive on the 2-arcs of Г. It has been shown that there exists a family of non-quasiprimitive （PSU3（q）, 2）-arc transitive graphs where q = 2^3m with m an odd integer. In this paper we investigate the case where q is an odd prime power.     

Groups with an element permuting with finitely many of its conjugates

We study the problem as to which is the cardinality of connected components of the graph Γ_α, defined as follows. Let G be a group and a an element of G. The vertex set V(Γ_α) of the graph is the conjugacy class of elements,Cl _G(a), and two vertices x and y of the graph Γ_α are bridged by an edge iff x=y. If the intersectionC _G(a)∩Cl _G(a) is finite, Γ_α is locally finite. We prove that connected components of the locally finite graph Γ_α are finite in some classes of groups. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 543–551, September–October, 1996.     

The higher lower central series

The Baer invariants Γ _n (G) of a group are central extensions of the elementsγ _n (G) of the lower central series. We show that the inclusionsγ _n +1 ⊂γ _n can be lifted to functor morphisms Γ _n+1→Γ _n and a canonical Lie algebra, analogous to Lazard’s Lie algebra, can be constructed which is explicitly computable. This is applied in various ways.     

Bijective maps on standard Borel subgroup of symplectic group preserving commutators

Let F be a field, and let \mathbbG\mathbb{G} be the standard Borel subgroup of the symplectic group Sp(2m, F). In this paper, we characterize the bijective maps ϕ: \mathbbG\mathbb{G} → \mathbbG\mathbb{G} satisfying ϕ[x, y] = [ϕ(x), ϕ(y)].     

Rigidity problem for lattices in solvable Lie groups

The paper concerns rigidity problem for lattices in simply connected solvable Lie groups. A lattice Γ⊂G is said to be rigid if for any isomorphism ϕ:Γ→Γ′ with another lattice Γ′⊂G there exists an automorphism which extends ϕ. An effective rigidity criterion is proved which generalizes well-known rigidity theorems due to Malcev and Saito. New examples of rigid and nonrigid lattices are constructed. In particular, we construct: a) rigid lattice Γ⊂G which is not Zariski dense in the adjoint representation ofG, b) Zariski dense lattice Γ⊂G which is not rigid, c) rigid virtually nilpotent lattice Γ in a solvable nonnilpotent Lie groupG.     

Groups containing an element that permutes with a finite number of its conjugates

We study a group G containing an element g such that C_G(g)∩g^G is finite. The nonoriented graph Γ is defined as follows. The vertex set of Γ is the conjugacy class g^G. Vertices x and y of the graph G are bridged by an edge iff x≠y and xy=yx. Let Γ₀ be some connected component of G. On a condition that any two vertices of Γ₀ generate a nilpotent group, it is proved that a subgroup generated by the vertex set of Γ₀ is locally nilpotent. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 637–650, November–December, 1998.     

Conjugacy classes of non-connected semisimple algebraic groups

Consider a non-connected algebraic group G = G ⋉ Γ with semisimple identity component G and a subgroup of its diagram automorphisms Γ. The identity component G acts on a fixed exterior component Gτ, id ≠ τ ∈ Γ by conjugation. In this paper we will describe the conjugacy classes and the invariant theory of this action. Let T be a τ -stable maximal torus of G and its Weyl group W. Then the quotient space Gτ//G is isomorphic to (T/(1 − τ )(T))/W^τ. Furthermore, exploiting the Jordan decomposition, the reduced fibres of this quotient map are naturally associated bundles over semisimple G-orbits. Similar to Steinberg's connected and simply connected case [22] and under additional assumptions on the fundamental group of G, a global section to this quotient map exists. The material presented here is a synopsis of the Ph.D thesis of the author, cf. [15].     

Dynamical properties of groups of automorphisms on Heisenberg nilmanifolds

Dynamical properties of actions of groups of automorphisms on Heisenberg nilmanifolds H/Γ are studied. It is proved that such a group G has only finite or dense orbits if the induced action on the associated torus has the same property. This gives a partial answer to a question of Margulis. Moreover, the G-invariant (or even stationary) measures on H/Γ are determined.     

Homological vanishing theorems for locally analytic representations

Let G be the group of rational points of a split connected reductive group over a p-adic local field, and let Γ be a discrete and cocompact subgroup of G. Motivated by questions on the cohomology of p-adic symmetric spaces, we investigate the homology of Γ with coefficients in locally analytic principal series and related representations of G. The vanishing and finiteness results that we find partially rely on the compactness of certain Banach–Hecke operators. We also give a new construction of P. Schneider’s reduced Hodge–de Rham spectral sequence and show that the induced filtration is the Hodge–de Rham filtration. In a previously unknown case, our vanishing theorems then also imply two other of P. Schneider’s conjectures.