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.     

Solvability of the conjugacy problem for finite subgroups in automorphism groups of free groups

We prove that there exists an algorithm which solves a conjugacy problem for finite subgroups in automorphism and outer automorphism groups of a free group of finite rank. Of independent interest is the construction of an algorithm of decomposing an arbitrary free-by-finite group into a fundamental group of a finite graph of finite groups, with the number of steps evaluated explicitly. In passing, we solve the conjugacy problem for finite subgroups in almost free groups. As a consequence, an algorithm is obtained computing generating sets for a group of fixed points in an arbitrary finite automorphism group of a free group of finite rank.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 558–606, September-October, 1995.Supported by the RFFR grant No. 93-011-1508 and by the ISF (International Science Foundation) grant RPC000.     

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).     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

On the Automorphism Group of a Free Pro-l Meta-abelian Group and an Application to Galois Representations

In § l of this article, we study group-theoretical properties of some automorphism group Ψ^* of the meta-abelian quotient § of a free pro-l group § of rank two, and show that the conjugacy class of some element of order two of Ψ^* is not determined by the action induced on the abelian quotient § of § in the case of § = 2. In § 2 we apply the results to the outer Galois representation § attached to the curve C deleted one point from an elliptic curve E, and give an example that §_c does not factor through the l-adic representation attached to E.     

Small maximal pro-p Galois groups

For an odd prime p we classify the pro-p groups of rank ≤ 4 which are realizable as the maximal pro-p Galois group of a field containing a primitive root of unity of order p. Received: 2 September 1997     

Solvable Subgroups of Out(F n ) are Virtually Abelian

Let F _n be the free group of rank n, let Aut(F _n ) be its automorphism group and let Out(F _n ) be its outer automorphism group. We show that every solvable subgroup of Out(F _n ) has a finite index subgroup that is finitely generated and free Abelian. We also show that every Abelian subgroup of Out(F _n ) has a finite index subgroup that lifts to Aut(F _n ).     

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp ² possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.     

A Classification of Fully Residually Free Groups of Rank Three or Less

A groupGisfully residually freeprovided to every finite setS ⊂ G\{1} of non-trivial elements ofGthere is a free groupF_Sand an epimorphismh_S:G → F_Ssuch thath_S(g) ≠ 1 for allg ∈ S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.     

Virtually free pro-p groups

We prove that in the category of pro-p groups any finitely generated group G with a free open subgroup splits either as an amalgamated free product or as an HNN-extension over a finite p-group. From this result we deduce that such a pro-p group is the pro-p completion of a fundamental group of a finite graph of finite p-groups.     

Free-by-finite pro-p groups and integral p-adic representations

Let F be a free pro-p group of finite rank n and C_p^r{C_{p^r}} a cyclic group of order p ^r . In this work we classify p-adic representations C_p^r? GL_n(\mathbbZ_p){ C_{p^r}\longrightarrow GL_n(\mathbb{Z}_{p})} that can be obtained as a composite of an embedding C_p^r? Aut(F){C_{p^r}\longrightarrow {\rm Aut}(F)} with the natural epimorphism Aut(F)? GL_n(\mathbbZ_p){{\rm Aut}(F)\longrightarrow GL_n(\mathbb{Z}_{p})} .     

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.     

Geometry for palindromic automorphism groups of free groups

We examine the palindromic automorphism group , of a free group F _n , a group first defined by Collins in [5] which is related to hyperelliptic involutions of mapping class groups, congruence subgroups of , and symmetric automorphism groups of free groups. Cohomological properties of the group are explored by looking at a contractible space on which acts properly with finite quotient. Our results answer some conjectures of Collins and provide a few striking results about the cohomology of , such as that its rational cohomology is zero at the vcd. Received: January 17, 2000.     

Kirillov's Orbit Method for p-Groups and Pro-p Groups

In this text, we study Kirillov's orbit method in the context of Lazard's p-saturable groups when p is an odd prime. Using this approach we prove that the orbit method works in the following cases: torsion free p-adic analytic pro-p groups of dimension smaller than p, pro-p Sylow subgroups of classical groups over ? _p of small dimension and for certain families of finite p-groups.     

Automorphism groups of graphs with forbidden subgraphs

For a graph Ф letF(Ф) be the class of finite graphs which do not contain an induced subgraph isomorphic to Ф. We show that whenever Ф is not isomorphic to a path on at most 4 vertices or to the complement of such a graph then for every finite groupG there exists a graph ГєF(Ф) such thatG is isomorphic to the automorphism group of Г. For all paths д on at most 4 vertices we determine the class of all automorphism groups of members ofF(д).     

A census of semisymmetric cubic graphs on up to 768 vertices

A list is given of all semisymmetric (edge- but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3,3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K_1,3 as a normal quotient is investigated in detail. Supported in part by N.Z. Marsden Fund (grant no. UOA 124) and N.Z. Centres of Research Excellence Fund (grant no. UOA 201) Supported in part by “Ministrstvo za šolstvo, znanost in šport Slovenije”, research program no. 101-506. Supported in part by research projects no. Z1-4186-0101 and no. Z1-3124-0101. The fourth author would like to thank the University of Auckland for hospitality during his visit there in 2003.     

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.     

Counting finite index subgroups and the P. Hall enumeration principle

We apply the P. Hall enumeration principle to count the number of subgroups of a given index in the free pro-p group and the free abelian group. We shall present an infinite family of non-isomorphic pro-p groups with the same zeta function.