On homogeneous connections with exotic holonomy

In Proc. Symp. Pure Math. 53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H ₃-connections (G ₃-connections resp.).In this paper, we give a complete classification of homogeneous G ₃-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G ₃-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G ₃-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc. 345 (1994), 293–321) which states that there are no compact manifolds with an H ₃-connection.     

A characterization of powerful p-groups

In [10] Benjamin Klopsch and Ilir Snopce recently posted the conjecture that for p ≥ 3 and G a torsion-free pro-p group, d(G) = dim(G) is a sufficient and necessary condition for the pro-p group G to be uniform. They pointed out that this follows from the more general question of whether for a finite p-group d(G) = log _p (|Ω₁(G)|) is a sufficient and necessary condition for the group G to be powerful. In this short note we will give a positive answer to this question for p ≥ 5.     

Warfield’s Duality and Malcev’s Matrix

In this work, we investigate relations between Malcev’s matrices of a torsion-free group G of finite rank and Malcev’s matrices of groups Hom(R,G) and Hom(G,R), where G is a locally free group and R is a torsion-free group of rank 1.     

Malnormal subgroups of lattices and the Pukánszky invariant in group factors

Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A. This implies that the II₁ factor VN(Γ) contains a masa A with Pukánszky invariant {∞}.     

Holomorphic representations of nilpotent Lie groups

Let G be a connected, simply-connected complex nilpotent Lie group, and G_r ?( G a real form of G. Motivated by the problem of analytic continuation of Banach-space representations of G_R to holomorphic representations of G, we construct translation-invariant locally-convex algebras of entire functions on G (generalizing the classical spaces of entire functions of finite exponential order). The dual spaces of these algebras are naturally identified with algebras of left-invariant differential operators of infinite order on G. In connection with analytic continuation of unitary representations of G_R, we study the convex cone of entire functions on G whose restrictions to G_R are positive-definite, and determine the minimal order of growth at infinity of such functions.     

The topology of moduli spaces of group representations: The case of compact surface

Let G be a connected complex semisimple affine algebraic group, and let K be a maximal compact subgroup of G. Let X be a noncompact oriented surface. The main theorem of Florentino and Lawton (2009) [3] says that the moduli space of flat K-connections on X is a strong deformation retraction of the moduli space of flat G-connections on X. We prove that this statement fails whenever X is compact of genus at least two.     

Dimensions of division rings

Letk be a field. WriteD(G) for the quotient division ring of the group ringkG of a torsion-free, polycyclic-by-finite groupG, andD(g) for the quotient ring of the enveloping algebra of a finite-dimensional Lie algebrag overk. In this note we show that the Hirsch numberh(G) and dim _k g are invariants for the respective division rings, by calculating the Krull and global dimensions ofD(G)? _k D(G) andD(g)? _k D(g).     

Absolute Nil-Ideals of Abelian Groups

It is known that in an Abelian group G that contains no nonzero divisible torsion-free subgroups the intersection of upper nil-radicals of all the rings on G is $\bigcap\limits_{p} pT(G)$ , where T(G) is the torsion part of G. In this work, we define a pure fully invariant subgroup G*???T(G) of an arbitrary Abelian mixed group G and prove that if G contains no nonzero torsion-free subgroups, then the subgroup $\bigcap\limits_{p} pG^{*}$ is a nil-ideal in any ring on G, and the first Ulm subgroup G1 is its nilpotent ideal.     

On Degrees of Growth of Finitely Generated Groups

We prove that for an arbitrary function ρ of subexponential growth there exists a group G of intermediate growth whose growth function satisfies the inequality v _G,S (n) ? ρ(n) for all n. For every prime p, one can take G to be a p-group; one can also take a torsion-free group G. We also discuss some generalizations of this assertion.     

Almost fixed-point-free automorphisms of soluble groups

Suppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m=4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].     

A descent principle for the Dirac-dual-Dirac method

Let G be a torsion-free discrete group with a finite-dimensional classifying space BG. We show that G has a dual-Dirac morphism if and only if a certain coarse (co-)assembly map is an isomorphism. Hence the existence of a dual-Dirac morphism for such groups is a metric, that is, coarse, invariant. We get results for groups with torsion as well.     

Graphs without K4 and well-quasi-ordering

It is proved that given an infinite sequence G₁, G₂, G₃,…, of series-parallel graphs there are indices i < j such that G_j contains an induced subgraph contractable onto G_i. An example is given showing that for planar graphs the preceding theorem fails.     

H-domination in graphs

Let H be some fixed graph of order p. For a given graph G and vertex set S⊆V(G), we say that S is H-decomposable if S can be partitioned as S=S₁∪S₂∪?∪S_j where, for each of the disjoint subsets S_i, with 1?i?j, we have |S_i|=p and H is a spanning subgraph of 〈S_i〉, the subgraph induced by S_i. We define the H-domination number of G, denoted as γ_H(G), to be the minimum cardinality of an H-decomposable dominating set S. If no such dominating set exists, we write γ_H(G)=∞. We show that the associated H-domination decision problem is NP-complete for every choice of H. Bounds are shown for γ_H(G). We show, in particular, that if δ(G)?2, then γ_P3(G)?3γ(G). Also, if γ_P3(G)=3γ(G), then every γ(G)-set is an efficient dominating set.     

Matroids over fp which are rational excluded minors

For a given prime p, we construct a collection of 2^p matroids G_p,a with (1) χ_pf(G_p,a)={p}, and (2) G_p,a is an excluded minor for rational representability. The motivating construction (Section 2) disproves a conjectures of Reid [4], using relatively high-rank, high cardinality matroids. The general construction (Section 3) makes use of ordered partitions (χ_pf(G) denotes the prime-field characteristic set of G, i.e., the set of prime fields over which G may be represented, while G can be represented over fields of no other characteristic.) Finally, Section 4 offers another construction with the same properties–a kind of projective dual to Section 2.     

On Dominions of the Rationals in Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a₁,..., a_n) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.     

Quasi-co-local subgroups of abelian groups

Let {0}≠K be a subgroup of the abelian group G. In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 29-37], K was called a co-local (cl) subgroup of G if is naturally isomorphic to . We generalize this notion to the quasi-category of abelian groups and call the subgroup K≠{0} of G a quasi-co-local (qcl) subgroup of G if is naturally isomorphic to . We show that qcl subgroups behave quite differently from cl subgroups. For example, while cl subgroups K are pure in G, i.e. G/K is torsion-free if G is torsion-free, any reduced torsion group T can be the torsion subgroup t(G/K) of G/K where G is torsion-free and K is a qcl subgroup of G.     

Uniform K-homology theory

We define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C^∗-algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C^∗-algebras. We show that our theory has a Mayer-Vietoris sequence. We prove that for a torsion-free countable discrete group Γ, the direct limit of the uniform K-homology of the Rips complexes of Γ, , is isomorphic to , the left-hand side of the Baum-Connes conjecture with coefficients in ?^∞Γ. In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a “fundamental class”, in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras.