SPECIAL REDUCTIVE GROUPS OVER AN ARBITRARY FIELD

A linear algebraic group G defined over a field k is called special if every G-torsor over every field extension of k is trivial. In 1958 Grothendieck classified special groups in the case where the base field is algebraically closed. In this paper we describe the derived subgroup and the coradical of a special reductive group over an arbitrary field k. We also classify special semisimple groups, special reductive groups of inner type, and special quasisplit reductive groups over an arbitrary field k. Finally, we give an application to a conjecture of Serre.     

Cartesian Closed Subcategories of $CONT_{\ll }^{{\kern 8pt}\ast }$

Let CONT _? be the category of continuous domains and Scott continuous mappings that preserve the way-below relation on domains. Let ω-ALG _? be the full subcategory of CONT _? consisting of all countably based algebraic domains, and F I N be the category of finite posets and monotone mappings. The main result proved in this paper is that F I N is the largest Cartesian closed full subcategory of ω-ALG _?. On the other hand, it is shown that the algebraic L-domains form a Cartesian closed full subcategory of ALG _?.     

Degeneration of Horospheres in Spherical Homogeneous Spaces

Horospheres for an action of a semisimple algebraic group G on an affine variety X are the generic orbits of a maximal unipotent subgroup U ? G or, equivalently, the generic fibers of the categorical quotient of the variety X by the action of U, which is defined by the values of the highest weight functions. The remaining fibers of this quotient (which we call degenerate horospheres) for a certain class of spherical G-varieties containing all simply connected symmetric spaces are studied.     

On locally finite Cpp-groups

A group G is called a Cpp-group for a prime number p, if G has elements of order p and the centralizer of every non-trivial p-element of G is a pgroup. In this paper we prove that the only infinite locally finite simple groups that are Cpp-groups are isomorphic either to PSL(2,K) or, if p = 2, to Sz(K), with K a suitable algebraic field over GF(p). Using this fact, we also give some structure theorems for infinite locally finite Cpp-groups.     

The Schwartz space of a smooth semi-algebraic stack

Schwartz functions, or measures, are defined on any smooth semi-algebraic (“Nash”) manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are defined as geometric stacks in the category of Nash manifolds. Moreover, when those are obtained from algebraic quotient stacks of the form X/G, with X a smooth affine variety and G a reductive group defined over a number field k, we define, whenever possible, an “evaluation map” at each semisimple k-point of the stack, without using truncation methods. This corresponds to a regularization of the sum of those orbital integrals whose semisimple part corresponds to the chosen k-point. These evaluation maps produce, in principle, a distribution which generalizes the Arthur–Selberg trace formula and Jacquet’s relative trace formula, although the former, and many instances of the latter, cannot actually be defined by the purely geometric methods of this paper. In any case, the stack-theoretic point of view provides an explanation for the pure inner forms that appear in many versions of the Langlands, and relative Langlands, conjectures.     

On the character of certain tilting modules

Let G be a semisimple group over an algebraically closed field of characteristic p > 0. We give a (partly conjectural) closed formula for the character of many indecomposable tilting rational G-modules assuming that p is large.     

Examples of algebraic groups of type <Emphasis Type="Italic">G</Emphasis> <Subscript>2</Subscript> having the same maximal tori

Answering a question of A. Rapinchuk, we construct examples of non-isomorphic semisimple algebraic groups H₁ and H₂ of type G₂ having coherently equivalent systems of maximal k-tori.     

On finite simple images of triangle groups

For a simple algebraic group G in characteristic p, a triple (a, b, c) of positive integers is said to be rigid for G if the dimensions of the subvarieties of G of elements of order dividing a, b, c sum to 2 dim G. In this paper we complete the proof of a conjecture of the third author, that for a rigid triple (a, b, c) for G with p > 0, the triangle group T_a,b,c has only finitely many simple images of the form G(p^r). We also obtain further results on the more general form of the conjecture, where the images G(p^r) can be arbitrary quasisimple groups of type G.     

Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups

Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S³ = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S³ of all geodesics on (G, d) with respect to p.     

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.     

Principal blocks and p-radical groups

Let G be a finite group and k a field of characteristic p > 0. In this paper, we obtain several equivalent conditions to determine whether the principal block B₀ of a finite p-solvable group G is p-radical, which means that B₀ has the property that e₀(k_P)^G is semisimple as a kG-module, where P is a Sylow p-subgroup of G, k_P is the trivial kP-module, (k_P)^G is the induced module, and e₀ is the block idempotent of B₀. We also give the complete classification of a finite p-solvable group G which has not more than three simple B₀-modules where B₀ is p-radical.     

Characterization of the alternating groups by their order and one conjugacy class length

Let G be a finite group, and let N(G) be the set of conjugacy class sizes of G. By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N(G) = N(L), then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In this article, we investigate validity of Thompson’s conjecture under a weak condition for the alternating groups of degrees p+1 and p+2, where p is a prime number. This work implies that Thompson’s conjecture holds for the alternating groups of degree p + 1 and p + 2.     

QUANDLE VARIETIES,GENERALIZED SYMMETRIC SPACES,AND <Emphasis Type="Italic">φ</Emphasis>-SPACES

We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ?. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry.Assume that Q is normal as an algebraic variety and that the action of its inner automorphism group Inn(Q) has a dense orbit. Then we show that there is an algebraic group G acting on Q with the same orbits as Inn(Q) such that each G-orbit is isomorphic to the quandle (G/H, ?φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.     

Rigid actions of amenable groups

Given a countable discrete amenable group G, does there exist a free action of G on a Lebesgue probability space which is both rigid and weakly mixing? The answer to this question is positive if G is abelian. An affirmative answer is given in this paper, in the case that G is solvable or residually finite. For a locally finite group, the question is reduced to an algebraic one. It is exemplified how the algebraic question can be positively resolved for some groups, whereas for others the algebraic viewpoint suggests the answer may be negative.     

On the second cohomology of an algebraic group and of its lie algebra in a positive characteristic

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h ? 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL₂(k), SL₃(k), SL₄(k), Sp₄(k), and G ₂, nontrivial isomorphisms of this kind exist for SL₄(k) and G ₂ only. For SL₄(k), there are two simple modules with nontrivial second cohomology and, for G ₂, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.     

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.     

Properly discontinuous group actions on affine homogeneous spaces

Let G be a real algebraic group, H ≤ G an algebraic subgroup containing a maximal reductive subgroup of G, and Γ a subgroup of G acting on G/H by left translations. We conjecture that Γ is virtually solvable provided its action on G/H is properly discontinuous and ΓG/H is compact, and we confirm this conjecture when G does not contain simple algebraic subgroups of rank ≥2. If the action of Γ on G/H (which is isomorphic to an affine linear space Aⁿ) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for n ≤ 5.     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

On <Emphasis Type="Italic">c</Emphasis><Superscript>#</Superscript>-normal subgroups infinite groups

A subgroup H of a finite group G is called a c^#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is a CAP-subgroup of G: In this paper, we investigate the influence of fewer c^#-normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.     

Semisimple weakly symmetric pseudo-Riemannian manifolds

We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature \((n-1,1)\) and trans-Lorentzian signature \((n-2,2)\).