Equations and algebraic geometry over profinite groups

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

La structure profinie du groupe des unités d'un anneau de séries entières à coefficients dans un anneau fini

The following theorem is proved: Let T be an uncountable set of algebraically independent elements over a field Ko. Then K = KO(T) is a Hilbertian field but the set of a E G( K) for which K(a) is PAC is nonmeasurable. Introduction. A field M is said to be pseudo algebraically closed (= PAC) if every nonempty absolutely irreducible variety V defined over M has an M-rational point. If M is an algebraic extension of a field K and every absolutely irreducible polynomial f E K [ X, Y], separable in Y, has infinitely many M-rational zeros, then M is PAC. This is a combination of Ax's application of descent [1] and the generic hyperplane intersection method as in Frey [3]. If a,,...,ae are e-elements of the absolute Galois group G(K) of K, then K(a) denotes the fixed field in K of a1,. . . ,0e. Here K is the algebraic closure of K. We denote by 4 the normalized Haar measure of G(K )e. It is proved in [6, Lemma 2.4] that if K is a Hilbertian field, if f E K[X, Y] is an absolutely irreducible polynomial and if A(f) = (a E G(K)e If has a K(a)-zero), then u(A(f )) = 1. If in addition K is countable, then there are only countably many f 's and therefore the intersection of all the A( f )'s is also a set of measure 1. Thus the set Se(K) = (a E G(K )e I K(a) is PAC} has measure 1. This basic result, which is called the Nullstellensatz in [6], has been the cornerstone for several model theoretic investigations of the fields K( a) (cf. [9, 7 and 4]). If K is uncountable, then the above argument is not valid any more. It is our aim in this note to show that indeed the Nullstellensatz itself is not true in this case. More precisely, we prove THEOREM. Let T be an uncountable set of algebraically independent elements over a field Ko. Then K = KO T) is a Hilbertian field but Se( K) is a nonmeasurable subset of G( K )e for every positive integer e. 1. The Haar measure of a profinite group. Let G be a profinite group and consider the boolean algebra of open-closed sets in G. They are finite unions of left cosets xN, where N are open normal subgroups. The a-algebra generated by the open-closed sets is denoted by 6J0. Every open subset of G is a union of open-closed sets. We Received by the editors Februarv 5. 1982 and, in revised form, May 27, 1982. 1980 Mathematics Subject Clavsificcation. Primary 1 2F20. Kev words and phrases. PAC fields, Hilbertian fields, Haar measure. ' Partially supported by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities. i) 1983 American Mathematical Society 0002-9939/82/0000-0592/$02. 50

Let K be a non-archimedean local field and U a compact subgroup of GLn(K). Such U is a profinite group, actually it is virtually pro-p. If char(K) = 0, then U is finitely generated and even finitely presented as a profinite group. On the other hand if char(K) = p > 0, then U need not be finitely generated. Moreover, even if U is finitely generated, it may be non-finitely presented (see(1.3)). On the other hand;

Strongly generating elements in finite and profinite groups

In this expanded account of a talk given at the Oberwolfach Arbeitsgemeinschaft Totally Disconnected Groups, October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological paying special attention to the class of finitely generated profinite groups. Our primary source is a paper, entitled Generators and commutators in finite groups; abstract quotients of compact groups, that was published by Nikolov and Segal in 2012. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject.

We study profinite groups which are infinitely iterated wreath products W ∞ = …≀C n 2 ≀C n 1 of finite cyclic groups via combinatorial language of transducers. Namely, we provide a naturally defined automaton realization of the group W ∞ by an automaton over a changing alphabet. Our construction gives a characterization of these profinite groups as automaton groups, i.e. as groups generated by a single automaton.

We develop an approach to the character theory of certain classes of finite and profinite groups based on the construction of a Lie algebra associated to such a group, but without making use of the notion of a polarization which is central to the classical orbit method. Instead, Kirillov’s character formula becomes the fundamental object of study. Our results are then used to produce an alternate proof of the orbit method classification of complex irreducible representations of p-groups of nilpotence class < p, where p is a prime, and of continuous complex irreducible representations of uniformly powerful pro-p-groups (with a certain modification for p = 2). As a main application, we give a quick and transparent proof of the p-adic analogue of Brown’s theorem, stating that for a nilpotent Lie group over ℚp the Fell topology on the set of isomorphism classes of its irreducible representations coincides with the quotient topology on the set of its coadjoint orbits.

A group L is primitive monolithic if L has a unique minimal normal subgroup, N, and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L, given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N. Information about PL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.

A group will be called full if it is the Galois group of an algebraic closure of a field. In this paper we first investigate full Abelian groups and classify them. Then we examine full groups from the point of view of how we can operate on them and still maintain the property of being full. Of course, by the fundamental theorem of the infinite Galois theory, closed subgroups (with standard profinite topology) of full groups are full. In general, products of full groups are notfull (for example, Z2 × Z2 is not full, by a theorem of Artin and Schreier (1927)); however we produce a set of groups which can always be attached as direct factors to full groups and still retain full groups. For the definition and basic properties of profinite groups we refer the reader to Cassels and Frohlich (1967). It has been shown by Leptin (1955) and independently by the author that profinite groups are Galois groups. The author (1971) has shown that if G is profinite, then G = Gdl(k/L) where L may have any desired characteristic and contains all primitive nth roots of unity, for all n. We will denote by p the pro-p-group which is the inverse limit of cyclic p-groups. (This is the group of p-adic integers).

The object of study of this paper is the following multi-determinantal algebraic variety, $\text{SING}_{n, m}$ , which captures the symbolic determinant identity testing (SDIT) problem (a canonical version of the polynomial identity testing (PIT) problem), and plays a central role in algebra, algebraic geometry and computational complexity theory. $\text{SING}_{n, m}$ is the set of all $m$ -tuples of $n\times n$ complex matrices which span only singular matrices. In other words, the determinant of any linear combination of the matrices in such a tuple vanishes. The algorithmic complexity of testing membership in $\text{SING}_{n, m}$ is a central question in computational complexity. Having almost a trivial probabilistic algorithm, finding an efficient deterministic algorithm is a holy grail of derandomization, and to top it, will imply super-polynomial circuit lower bounds! A sequence of recent works suggests efficient deterministic “geodesic descent” algorithms for memberships in a general class of algebraic varieties, namely the null cones of (reductive) linear group actions. Can such algorithms be used for the problem above? Our main result is negative: $\text{SING}_{n, m}$ is not the null cone of any such group action! This stands in stark contrast to a non-commutative analog of this variety (for which such algorithms work), and points to an inherent structural difficulty of $\text{SING}_{n, m}$ . In other words, we provide a barrier for the attempts of derandomizing SDIT via these algorithms. To prove this result we identify precisely the group of symmetries of $\text{SING}_{n, m}$ . We find this characterization, and the tools we introduce to prove it, of independent interest. Our characterization significantly generalizes a result of Frobenius for the special case $m=1$ (namely, computing the symmetries of the determinant). Our proof suggests a general method for determining the symmetries of general algebraic varieties, an algorithmic problem that was hardly studied and we believe is central to algebraic complexity.

We investigate p‐adic completions of clopen (i.e., closed and open at the same time) subgroups W of loop groups and diffeomorphism groups G of compact manifolds over non‐Archimedean fields. We outline two different compactifications of loop groups and one compactification of diffeomorphism groups, describe associated finite groups in projective limits, and discuss relations with the representation theory.

In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.

In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard's mathematics department was at the center of these developments. A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics--in algebraic geometry, complex analysis, and other esoteric subdisciplines that are rarely written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose. The story begins in 1825, when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics--an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce's successors transformed the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff. Influential figures soon flocked to Harvard, some overcoming great challenges to pursue their elected calling. A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond.