Finitely Generated Groups, p-Adic Analytic Groups and Poincare Series

Zeta functions enumerating normal subgroups of [formula omitted]-groups and their behavior on residue classes

A Weyl group multiple Dirichlet seriesis a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank rof the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group Wof Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n th order Gauss sums attached to the root system of type A 2. The basic technique is that of [11, 10]; namely, we construct a rational function in rvariables invariant under a certain action of W, and use this to build a “local factor” of the global series.

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

Automorphism groups of nilpotent groups and spaces

Let [Formula: see text] be a division ring with center [Formula: see text] and multiplicative group [Formula: see text], of char [Formula: see text], and with an involution *. Let [Formula: see text] be the group of unitary units of [Formula: see text], namely [Formula: see text]. We investigate various instances where the dimension [Formula: see text], and in which every non-central normal subgroup [Formula: see text] contains a free non cyclic subgroup. Among them, we consider the cases where [Formula: see text] is either generated over its center of characteristic 0 by a torsion free nilpotent group, or [Formula: see text] is the field of fractions of a group algebra [Formula: see text] of the residually torsion free nilpotent group [Formula: see text] over the field [Formula: see text] of characteristic 0 , or the field of fractions of the enveloping algebra of a locally solvable residually nilpotent Lie [Formula: see text]-algebra [Formula: see text].

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

In the classical theory of multiple zeta values (MZV’s), Furusho proposed a conjecture asserting that the p-adic MZV’s satisfy the same \({\mathbb {Q}}\)-linear relations that their corresponding real-valued MZV counterparts satisfy. In this paper, we verify a stronger version of a function field analogue of Furusho’s conjecture in the sense that we are able to deal with all linear relations over an algebraic closure of the given rational function field, not just the rational linear relations. To each tuple of positive integers \({\mathfrak {s}}=(s_1, \ldots , s_r)\), we construct a corresponding t-module together with a specific rational point. The fine resolution (via fiber coproduct) of this construction actually allows us to obtain nice logarithmic interpretations for both the \(\infty \)-adic MZV and v-adic MZV at \({\mathfrak {s}}\), completely generalizing the work of Anderson–Thakur (Ann Math (2) 132(1):159–191, 1990) in the case of \(r=1\). Furthermore it enables us to apply Yu’s sub-t-module theorem (Yu in Ann Math (2) 145(2):215–233, 1997), connecting any \(\infty \)-adic linear relation on MZV’s with a sub-t-module of a corresponding giant t-module. This makes it possible to arrive at the same linear relation for v-adic MZV’s.

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian) p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory.

The ‘degree of k-step nilpotence’ of a finite group G is the proportion of the tuples (x1,…, xk+1 ∈ Gk+1 for which the simple commutator [x1, …, xk+1] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Følner sequence if G is amenable. In our first main result we show that if G is finitely generated, then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent (Theorem 1.5). This generalises both an earlier result of the second author treating the case k = 1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ϕ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ϕ is sparse in a certain sense (Theorem 5.1). In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below, then G is virtually k-step nilpotent (Theorem 1.19), answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients (Theorem 1.21), generalising a result of Gallagher.

CHAPTER 4 - Recurrence Relations and Algebraic Equations

In the following chapter we address the techniques for the resolution of some celebrated recurrence relations. We will discuss in detail the linear recurrences with constant coefficients. Our emphasis goes to the application of the theory: the proofs, though elementary, are relegated to the end of the chapter. We proceed step by step in showing first how to solve just homogeneous recurrences, then how to find a particular solution in some special cases and only finally how to obtain all the solutions to the original problem. We also consider linear recurrences with variable coefficients and the divide and conquer recurrences: here we focus on the order of magnitude of the solutions, a fact which has an impact in the analysis of algorithms. There are about 40 examples and 50 classified problems.

A polynomial automorphism of a group G is an automorphism of the form . When G is polycyclic, we prove that the set PAut(G) of all polynomial automorphisms forms a polycyclic subgroup of the automorphism group Aut(G). When G is soluble, PAut(G) is not necessarily a subgroup, but we show that the subgroup generated by PAut(G) is soluble. We also give the general form of an element of PAut(G) when G is a torsion-free non-abelian metabelian nilpotent group.

Homotopy decompositions of polyhedra into products with the second factor [formula omitted

Fitting subgroups of profinite completions of soluble minimax groups