Equations and algebraic geometry over profinite groups

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.     

Normal Subgroups of Profinite Groups of Finite Cohomological Dimension

A profinite group G of finite cohomological dimension with (topologically)finitely generated closed normal subgroup N is studied. If Gis pro-p and N is either free as a pro-p group or a Poincarégroup of dimension 2 or analytic pro-p, it is shown that G/Nhas virtually finite cohomological dimension cd(G)–cd(N).Some other cases when G/N has virtually finite cohomologicaldimension are also considered. If G is profinite, the case of N projective or the profinitecompletion of the fundamental group of a compact surface isconsidered.     

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.     

Finite extensions of free pro-P groups of rank at most two

For a pro-p groupG, containing a free pro-p open normal subgroup of rank at most 2, a characterization as the fundamental group of a connected graph of cyclic groups of order at mostp, and an explicit list of all such groups with trivial center are given. It is shown that any automorphism of a free pro-p group of rank 2 of coprime finite order is induced by an automorphism of the Frattini factor groupF/F ^* . Finally, a complete list of automorphisms of finite order, up to conjugacy in Aut(F), is given. Supported by an NSERC grant. Supported by the Austrian Science Foundation.     

Test elements in pro-<Emphasis Type="Italic">p</Emphasis> groups with applications in discrete groups

Let G be a group. An element g ∈ G is called a test element of G if for every endomorphism ? : G → G, ?(g) = g implies ? is an automorphism. We prove that for a finitely generated profinite group G, g ∈ G is a test element of G if and only if it is not contained in a proper retract of G. Using this result we prove that an endomorphism of a free pro-p group of finite rank which preserves an automorphic orbit of a nontrivial element must be an automorphism. We give numerous explicit examples of test elements in free pro-p groups and Demushkin groups. By relating test elements in finitely generated residually finite-p Turner groups to test elements in their pro-p completions, we provide new examples of test elements in free discrete groups and surface groups. Moreover, we prove that the set of test elements of a free discrete group of finite rank is dense in the profinite topology.     

A virtually free pro-p need not be the fundamental group of a profinite graph of finite groups

A subgroup of a pro-p product with amalgamation of two p-groups is given which cannot be presented as the pro-p fundamental group of a profinite graph of p-groups.     

Homological Invariants for pro-p Groups and Some Finitely Presented pro- C{\cal C} Groups

Let G be a finitely presented pro- C{\cal C} group with discrete relations. We prove that the kernel of an epimorphism of G to [^(\Bbb Z)]_C\hat{\Bbb Z}_{\cal C} is topologically finitely generated if G does not contain a free pro- C{\cal C} group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]).For a pro-p group G of type FP_m we define a homological invariant C{\cal C} groups, pro-p groups, homological type FP_m, finite presentabilityBoth authors are partially supported by CNPq, Brazil.     

Generators of Simple Lie Algebras and the Lower Rank of Some Pro-P Groups

Abstract

Let  be a simple classical Lie algebra over a field F of characteristic p > 7. We show that > d () = 2, where d() is the number of generators of . Let G be a profinite group. We say that G has lower rank ≤ l, if there are {G _α} open subgroups which from a base for the topology at the identity and each G _α is generated (topologically) by no more than l elements. There is a standard way to associate a Lie algebra L(G) to a finitely generated (filtered) pro-p group G. Suppose L(G) ?  ? tF _p [t], where  is a simple Lie algebra over F _p , the field of p elements. We show that the lower rank of G is ≤ d () + 1. We also show that if  is simple classical of rank r and p > 7 or p 2r ² ? r, then the lower rank is actually 2.     

Virtually free factors of pro-p groups

Letp be a prime number,G a pro-p group, andH a closed (topologically) finitely generated subgroup ofG. We give conditions under whichH is virtually a free factor ofG, i.e., that there exists an open subgroupU ofG such thatU is the free pro-p product ofH and some other subgroup ofU. We prove that this happens if eitherG is a free pro-p group of any rank, or ifG is a free pro-p product of finitely generated pro-p groups. Research supported in part by grants from NSERC (Canada) and DGICYT (Spain).     

On the structure of compact torsion groups

It is shown that every profinite torsion group has a finite series of closed characteristic subgroups in which each factor either is a pro-p-group for some primep or is isomorphic to a Cartesian product of isomorphic finite simple groups.     

Procyclic coverings of commutators in profinite groups

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G′ such that the order of M is m-bounded and G′/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G′ is either finite of m-bounded order or procyclic.     

Homological Invariants for pro- p Grou ps and Some Finitely Presented pro-${\cal C}$ Grou ps

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]).For a pro-p group G of type FP_m we define a homological invariant ^m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FP_m. This generalises work of J. King for ¹(G) in the case when G is metabelian [9].Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on ¹(G) and on the automorphism group of G.Both authors are partially supported by CNPq, Brazil.     

Farrell cohomology and Brown theorems for profinite groups

LetG be a profinite group which has an open subgroupH such that the cohomologicalp-dimensiond≔cd_p(H) is finite (p is a fixed prime). The main result of this paper expresses thep-primary part of high degree cohomology ofG in terms of the elementary abelianp-subgroups ofG: From the latter one constructs a natural profinite simplicial setA G, on whichG acts by conjugation. ThenH ⁿ(G,M)≅H _G ⁿ (AG,M) holds forn≧d+r and everyp-primary discreteG-moduleM (r≔p-rank ofG). If one uses profinite Farrell cohomology, which is introduced in this paper, the analogous fact holds in all degrees. These results are the profinite analogues of theorems by K.S. Brown for discrete groups.     

On amenable groups of automorphisms on Von Neumann algebras

LetG ⊂ Aut ℳ be a countable group, ℳ a Von Neumann algebra. LetE be a set of pure states on ℳ such thatG*E ⊂E, S ^G be the set ofG invariant states on ℳ andS _E ^G =S ^G ∩w* cl coE. We investigate in this paper some geometric properties for the setS _E ^G which turn out to be equivalent to amenability for the groupG. For example, we show thatS _E ^G ⊂ ℳ_* (S _E ^G has the WRNP) implies that ℳ contains minimal projections (ê containsfinite G invariant orbits) hold true, for all ℳ iffG is amenable. Furthermore we show that ifG is amenable thenS ^G ∩M _* ^⊥ contains a big set, thus improving results obtained by Ching Chou in [2]. These results imply that no action of an amenable countable groupG on an arbitraryW* algebra ℳ iss — strongly ergodic. Moreover cardS ^G ∩M _* ^⊥ ≧2 ^c (see M. Choda [4], K. Schmidt [21] and compare with A. Connes and B. Weiss [5]). The author gratefully acknowledges the support of an Izaak Walton Killam Memorial Senior Fellowship.     

Virtually free pro-<Emphasis Type="Italic">p</Emphasis> products

It is shown that a finitely generated pro-p group G which is a virtually free pro-p product splits either as a free pro-p product with amalgamation or as a pro-p HNN-extension over a finite p-group. More precisely, G is the pro-p fundamental group of a finite graph of finitely generated pro-p groups with finite edge groups. This generalizes previous results of W. Herfort and the second author (cf. [2]).     

Groups with super-exponential subgroup growth

We show that, if the subgroup growth of a finitely generated (abstract or profinite) groupG is super-exponential, then every finite group occurs as a quotient of a finite index subgroup ofG. The proof involves techniques from finite permutation groups, and depends on the Classification of Finite Simple Groups.The first author was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T7441. The second author was partially supported by the Israeli National Science Foundation.     

Variations on polynomial subgroup growth

A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│ ^a subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ).     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

On some Λ-analytic pro-p groups

This paper is devoted to the first steps towards a systematic study of pro-p groups which are analytic over a commutative Noetherian local pro-p ring Λ, e.g. Λ= . We restrict our attention to Λ-standard groups, which are pro-p groups arising from a formal group defined over Λ. Under some additional assumptions we show that these groups are of ‘intermediate growth’ in various senses, strictly betweenp-adic analytic pro-p groups and free pro-p groups. This suggests a refinement of Lazard's theory which stresses the dichotomy betweenp-adic analytic pro-p groups and all the others. In the course of the discussion we answer a question posed in [LM1], and settle two conjectures from [Bo].     

On pro-p-groups with a single defining relator

We find necessary and sufficient conditions for the factor groups of the derived series of a pro-p-group with a single defining relation to be torsion free. For such groupsG we prove that the group algebra ℤ _pG is a domain and the cohomological dimension ofG is at most 2.