Pro-p completions of Poincaré duality groups

We consider some sufficient conditions for the pro-p completion of an orientable Poincaré duality group of dimension n ≥ 3 to be a virtually pro-p Poincaré duality group of dimension at most n ? 2.     

Finite automorphisms of negatively curved Poincaré duality groups

In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a -hyperbolic Poincaré Duality group over , then the fixed subgroup is a Poincaré Duality group over . We also provide a family of examples to show that the fixed subgroup might not be a Poincaré Duality group over . In fact, the fixed subgroups in our examples even fail to be duality groups over .     

Poincaré duality for p-adic Lie groups

We establish Poincaré duality for continuous group cohomology of p-adic Lie groups with rational coefficients and compare integral structures under this duality.     

Finiteness conditions on subgroups of profinite p-Poincaré duality groups

For a prime number p let G be a profinite p-PD _n group with a closed normal subgroup N such that G/N is a profinite p-PD _m group and that H _i (V, $ \mathbb{F} $ \mathbb{F} _p ) is finite for every open subgroup V of N and all i ≤ [n/2]. Generalising [12, Thm. 3.7.4] we show that m ≤ n and N is a profinite p-PD _{n − m} group. In case that G is a pro-p PD _n group of Euler characteristic 0 with a closed normal subgroup N of type FP _{[n−1 / 2]} such that G/N is soluble-by-finite pro-p group of finite rank, we show that N is a pro-p PD _{n − m} group, where m = vcd _p (G/N). As a corollary we obtain that a pro-p PD ₃ group with infinite abelianization is either soluble or contains a free nonprocyclic pro-p subgroup.     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.     

A geometric Schur–Weyl duality for quotients of affine Hecke algebras

After establishing a geometric Schur–Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts have nice ideals coming from geometry, allowing duality for quotients. Some of the quotients of the positive affine Hecke algebra are then identified to some cyclotomic Hecke algebras and the geometric setting allows the construction of canonical bases.     

Erratum to: “Solitary quotients of finite groups”

In this short note a correct proof of Theorem 3.3 from [T?rn?uceanu M., Solitary quotients of finite groups, Cent. Eur. J. Math., 2012, 10(2), 740–747] is given.     

Assouad–Nagata dimension of connected Lie groups

We prove that the asymptotic Assouad–Nagata dimension of a connected Lie group G equipped with a left-invariant Riemannian metric coincides with its topological dimension of G/C where C is a maximal compact subgroup. To prove it we will compute the Assouad–Nagata dimension of connected solvable Lie groups and semisimple Lie groups. As a consequence we show that the asymptotic Assouad–Nagata dimension of a polycyclic group equipped with a word metric is equal to its Hirsch length and that some wreath-type finitely generated groups can not be quasi-isometrically embedded into any cocompact lattice on a connected Lie group.     

Maximal resolvability of totally bounded and ℵ0-bounded groups

The maximal resolvability of totally bounded groups (and, under the assumption that the generalized continuum hypothesis holds, of ₀-bounded groups) is proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 593–598, April, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 94-01-01374.     

Classification of Poincaré duality algebras over the rationals

The Poincaré duality algebras over Q play a key role in the rational homotopy classification of closed manifolds [3]. In this paper we give a way of classifying general Poincaré duality algebras and then specialize to the case of algebras which are generated by some homogeneous component and show how the classification reduces to the linear classification of certain homogeneous polynomials and exterior forms.     

Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2–ℓ p minimization

The iteratively reweighted ? ₁ minimization algorithm (IRL1) has been widely used for variable selection, signal reconstruction and image processing. In this paper, we show that any sequence generated by the IRL1 is bounded and any accumulation point is a stationary point of the ? ₂–? _p minimization problem with 0<p<1. Moreover, the stationary point is a global minimizer and the convergence rate is approximately linear under certain conditions. We derive posteriori error bounds which can be used to construct practical stopping rules for the algorithm.     

Hyperbolic dimension and Poincaré critical exponent of rational maps

We study the Poincaré series of rational maps. By investigating the property of conical Julia set and dissipative measure, we prove that the Poincaré critical exponents are equal to the hyperbolic dimensions for a large class of rational maps.     

Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent {p > 2}

Let \({L(n)}\) be the language of group theory with n additional new constant symbols \({c_1,\ldots,c_n}\). In \({L(n)}\) we consider the class \({{\mathbb{K}}(n)}\) of all finite groups G of exponent \({p > 2}\), where \({G'\subseteq\langle c_1^G,\ldots,c_n^G\rangle \subseteq Z(G)}\) and \({c_1^G,\ldots,c_n^G}\) are linearly independent. Using amalgamation we show the existence of Fraïssé limits \({D(n)}\) of \({{\mathbb{K}}(n)}\). \({D(1)}\) is Felgner’s extra special p-group. The elementary theories of the \({D(n)}\) are supersimple of SU-rank 1. They have the independence property.     

Non-abelian composition factors of ℓ-Brauer m-rational groups

It is known that sporadic groups, alternating groups, and only finitely many groups of Lie type may occur as non-abelian composition factors of non-solvable m-rational groups. In this paper, we prove a similar statement for ?-Brauer m-rational groups, and give an explicit list of the possible non-abelian composition factors of ?-Brauer rational groups.     

On Selmer groups of abelian varieties over ℓ-adic Lie extensions of global function fields

Let F be a global function field of characteristic p > 0 and A/F an abelian variety. Let K/F be an ?-adic Lie extension (? ≠ p) unramified outside a finite set of primes S and such that Gal(K/F) has no elements of order ?. We shall prove that, under certain conditions, Sel _A (K) _? ^∨ has no nontrivial pseudo-null submodule.     

A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields

In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups Open image in new window for F a finite étale Galois module on K of order invertible in K and with \(F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))\). Furthermore, we prove that \(\mathrm {H}^1(K,G) = 0\) for G a simply connected, quasisplit semisimple group over K not of type \(E_8\).