Volume 21, Issue 4

Let æ : G ↪ GL( n , 𝔽) be a faithful representation of a finite group G . In this paper we study the image of the associated Noether map . It turns out that the image of the Noether map characterizes the ring of invariants in the sense that its integral closure . This is true without any restrictions on the group, representation, or ground field. Moreover, we show that the extension is a finite p -root extension if the characteristic of the ground field is p . Furthermore, we show that the Noether map is surjective, if V = 𝔽 n is a projective 𝔽 G -module. We apply these results and obtain upper bounds on the degrees of a minimal generating set of 𝔽[ V ] G and the Cohen-Macaulay defect of 𝔽[ V ] G . We illustrate our results with several examples.

We give a general definition of a subadditive invariant i of Mod( R ), where R is any ring, and the related notion of algebraic entropy of endomorphisms of R -modules, with respect to i . We examine the properties of the various entropies that arise in different circumstances. Then we focus on the rank-entropy, namely the entropy arising from the invariant ‘rank’ for Abelian groups. We show that the rank-entropy satisfies the Addition Theorem. We also provide a uniqueness theorem for the rank-entropy.

The maximal algebra of quotients of a semiprime Lie algebra was introduced recently by M. Siles Molina. In the present paper we answer some natural questions concerning this concept, and describe maximal algebras of quotients of certain Lie algebras that arise from associative algebras.

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lück-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of Mahler measures over finite groups, for example, in the classical case of the free abelian group or the infinite dihedral group, and others.

Let α be an infinite cardinal. Generalizing a recent result of Comfort and van Mill, we prove that every α -pseudocompact abelian group of weight > α has some proper dense α -pseudocompact subgroup and admits some strictly finer α -pseudocompact group topology.

Let FG be the group algebra of a group G without 2-elements over a field F of characteristic p ≠ 2 endowed with the canonical involution induced from the map g ↦ g –1 , g ∈ G . Let ( FG ) – and ( FG ) + be the sets of skew and symmetric elements of FG , respectively, and let P denote the set of p -elements of G (with P = 1 if p = 0). In the present paper we prove that if either P is finite or G is non-torsion and ( FG ) – or ( FG ) + is Lie solvable, then FG is Lie solvable. The remaining cases are also settled upon small restrictions.

We continue the investigations started in [Rivin, Technical Report math., 1999, Rivin, Duke Math. J., 2006]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Γ. We consider all walks of length N on G , starting from ν i and ending at ν j . To each such walk w we assign the element of Γ equal to the product of the elements along the walk. The set of all walks of length N from ν i to ν j thus induces a probability distribution F N,i,j on Γ. In [Rivin, Technical Report math., 1999] we give necessary and sufficient conditions for the limit as N goes to infinity of F N,i,j to exist and to be the uniform density on Γ (a detailed argument is presented in [Rivin, Duke Math. J., 2006]). The convergence speed is then exponential in N . In this paper we consider ( G , Γ), where Γ is a group possessing Kazhdan's property T (or, less restrictively, property τ with respect to representations with finite image), and a family of homomorphisms ψ k : Γ → Γ k with finite image. Each F N,i,j induces a distribution on Γ k (by push-forward under ψ k ). Our main result is that, under mild technical assumptions, the exponential rate of convergence of to the uniform distribution on Γ k does not depend on k . As an application, we prove effective versions of the results of [Rivin, Duke Math. J., 2006] on the probability that a random (in a suitable sense) element of SL( n , ℤ) or Sp( n , ℤ) has irreducible characteristic polynomial, generic Galois group, etc.

In this paper we consider Schrödinger operators H = –Δ + i ( A · ∇ + ∇ · A ) + V = –Δ + L in ℝ n , n ≥ 3. Under almost optimal conditions on A and V both in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not an immediate corollary of the free case as T ( λ ) := L (–Δ – ( λ 2 + i 0)) –1 is not small in operator norm on weighted L 2 spaces as λ → ∞. We instead deduce the existence of inverses ( I + T ( λ )) –1 by showing that the spectral radius of T ( λ ) decreases to zero. In particular, there is an integer m such that lim sup λ →∞ ∥ T ( λ ) m ∥ < . This is based on an angular decomposition of the free resolvent for which we establish the limiting absorption bound (0.1) ∥ D α ℛ d,δ ( λ 2 ) f ∥ B* ≤ C n λ –1+| α | ∥ f ∥ B where 0 ≤ | α | ≤ 2, B is the Agmon-Hörmander space, and ℛ d,δ ( λ 2 ) is the free resolvent operator at energy λ 2 whose kernel is restricted in angle to a cone of size δ and by d away from the diagonal x = y . The main point is that C n only depends on the dimension, but not on the various cut-offs. The proof of (0.1) avoids the Fourier transform and instead uses Hörmander's variable coefficient Plancherel theorem for oscillatory integrals.

We establish sufficient conditions for the nerve of the centric linking system of a p -local finite group ( S , ℱ, ℒ) to have the homotopy type of an Eilenberg-MacLane space K (Γ, 1) for a group Γ which contains S . We prove that in this situation the entire p -local finite group can be reconstructed from Γ. Our theorem applies to many of the known examples of exotic p -local finite groups.