The profinite polynomial automorphism group

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.

Let p , l be distinct prime numbers. A tripod-degree over p at l is defined to be an l -adic unit obtained by forming the image, by the l -adic cyclotomic character, of some continuous automorphism of the geometrically pro- l fundamental group of a split tripod over a finite field of characteristic p . The notion of a tripod-degree plays an important role in the study of the geometrically pro- l anabelian geometry of hyperbolic curves over finite fields, e.g., in the theory of cuspidalizations of the geometrically pro- l fundamental groups of hyperbolic curves over finite fields. In the present paper, we study the tripod-degrees. In particular, we prove that, under a certain condition, the group of tripod-degrees over p at l coincides with the closed subgroup of the group of l -adic units topologically generated by p . As an application of this result, we also conclude that, under a certain condition, the natural homomorphism from the group of automorphisms of the split tripod to the group of outer continuous automorphisms of the geometrically pro- l fundamental group of the split tripod that lie over the identity automorphism of the absolute Galois group of the basefield is surjective.

Let Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq¯ be a subfield of Fq. From any two q¯-linearized polynomials L1,L2∈F‾q[T] of degree q, we construct an ordinary curve X(L1,L2) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(L1,L2) over the algebraic closure F‾q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q¯−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(L1,L2)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(L1,L1))=Σ⋊Δ with a dihedral group Δ of order 2(q¯−1) containing Γ. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F‾q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(L1,L2) for some separable q¯-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin–Mumford curves, due to Arakelian and Korchmáros [1].

Let [Formula: see text], [Formula: see text] be the set of all [Formula: see text] matrices over a finite field [Formula: see text], and [Formula: see text] the subset of [Formula: see text] consisting of all rank one matrices. In this paper, we first determine the automorphism group and the fixing number of the orthogonality graph of [Formula: see text], and then characterize the automorphism group and the fixing number of the orthogonality graph of [Formula: see text].

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

Inserted in the context of algebraic curves defined over finite fields, this work presents several results in two different topics. First, it gives a complete characterization of the F q 3 -Frobenius nonclassical curves of type Y q 2 +q+1 = f (X), and it provides an explicit computation of the following birational invariants: genus, automorphism group, Hasse-Witt invariant and a-number. The number of F q 3 -rational points is computed as well. Second, this work provides an extensive study of the Hasse-Witt invariant of the curves Y m + X n + 1 = 0 and Y m + X n + X = 0. A combinatorial formula for this invariant is presented in the general case, and explicit closed formulas are provided for special values of m and n.

We investigate some connections between the Fraiss? theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ?-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements that are moved (not fixed) by any non-identity automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of an infinite sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. We note connections of the subject to computational group theory, asymptotic group theory, highly regular structures, and the Graph Isomorphism problem, and list a number of open 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.

We develop a new method to investigate groups acting on rooted trees. Considering the set of continuous functions from the automorphism group $\operatorname{Aut}X^*$ of a regular rooted tree to a finite field, we associate to each subgroup $G\leq\operatorname{Aut}X^*$ the ideal of functions which vanish on $G$. Our approach yields many new results for discrete groups as well as for closed subgroups of $\operatorname{Aut}X^*$ and some specific subgroups thereof. We exemplify the fruitfulness of this simple idea in many ways, including central series, automorphism towers, Hausdorff dimension, congruence problems, and a functorial construction of Lie algebras via affine group schemes.

In this paper, we prove that the etale fundamental group of a hyperbolic curve over an arithmetic field [e.g., a finite extension field of Q or Qp] or an algebraically closed field is indecomposable [i.e., cannot be decomposed into the direct product of nontrivial profinite groups]. Moreover, in the case of characteristic zero, we also prove that the etale fundamental group of the configuration space of a curve of the above type is indecomposable. Finally, we consider the topic of indecomposability in the context of the comparison of the absolute Galois group of Q with the Grothendieck-Teichmuller group GT and pose the question: Is GT indecomposable? We give an affirmative answer to a pro-l version of this question

Let F=(F_1,...,F_n):C^n --> C^n be a polynomial mapping. By the multidegree of the mapping F we mean mdeg F=(deg F_1,...,deg F_n), an element of N^n. The aim of this paper is to study the following problem (especially for n=3): for which sequence (d_1,...,d_n) in N^n there is a tame automorphism F of C^n such that mdeg F=(d_1,...,d_n). In other words we investigate the set mdeg(Tame(C^n)), where Tame(C^n) denotes the group of tame automorphisms of C^n and mdeg denotes the mapping from the set of polynomial endomorphisms of C^n into the set N^n. Since for all permutation s of {1,...,n} we have (d_1,...,d_n) is in mdeg(Tame(C^n)) if and only if (d_s(1),...,d_s(n)) is in mdeg(Tame(C^n)) we may focus on the set mdeg(Tame(C^n)) intersected with {(d_1,...,d_n) : d_1<=...<=d_n}. In the paper, among other things, we give complete description of the sets: mdeg(Tame(C^n)) intersected with {(3,d_2,d_3):3<=d_2<=d_3}}, mdeg(Tame(C^n)) intersected with {(5,d_2,d_3):5<=d_2<=d_3}}, In the examination of the last set the most difficult part is to prove that (5,6,9) is not in mdeg(Tame(C^n)). As a surprising consequence of the method used in proving that (5,6,9) is not in mdeg(Tame(C^n)), we obtain the result saying that the existence of tame automorphism F of C^3 with mdeg F=(37,70,105) implies that two dimensional Jacobian Conjecture is not true. Also, we give the complete description of the following sets: mdeg(Tame(C^n)) intersected with {(p_1,p_2,d_3):3<=p_1<p_2<=d_3}}, where p_1 and p_2 are prime numbers, mdeg(Tame(C^n)) intersected with {(d_1,d_2,d_3):d_1<p_2<=d_3}}, where d_1 and d_2 are odd numbers such that gcd(d_1,d_2)=1. Using description of the last set we show that the set mdeg(Aute(C^n))\mdeg(Tame(C^n)) is infinite.

On the number of curves of genus 2 over a finite field

Equivariant Poincaré Polynomials and Counting Points over Finite Fields

A tournament T is called symmetric if its automorphism group is transitive on the points and arcs of T. The main result of this paper is that if T is a finite symmetric tournament then T is isomorphic to one of the quadratic residue tournaments formed on the points of a finite field GF(pn), pn ≡ 3 (4), by the following rule: If a, b ∈ GF(pn) then there is an are directed from a to b exactly when b – a is a non-zero quadratic residue in GF(pn).