Complex Tori, Theta Groups and Their Jordan Properties (original) (raw)

Theta Groups and Products of Abelian and Rational Varieties

Proceedings of the Edinburgh Mathematical Society, 2013

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups does not hold for the groups of birational automorphisms of products of an elliptic curve and the projective line. This gives a negative answer to a question of V.

A characterization of compact complex tori via automorphism groups

Mathematische Annalen, 2013

We show that a compact Kähler manifold X is a complex torus if both the continuous part and discrete part of some automorphism group G of X are infinite groups, unless X is bimeromorphic to a non-trivial G-equivariant fibration. Some applications to dynamics are given. Theorem 1.1. Let X be a compact Kähler manifold and G ≤ Aut(X) a subgroup of automorphisms. Assume the following three conditions. (1) G 0 := G ∩ Aut 0 (X) is infinite.

Jordan property for non-linear algebraic groups and projective varieties

American Journal of Mathematics, 2018

A century ago, Camille Jordan proved that the complex general linear group GL n (C) has the Jordan property: there is a Jordan constant C n such that every finite subgroup H ≤ GL n (C) has an abelian subgroup H 1 of index [H : H 1 ] ≤ C n. We show that every connected algebraic group G (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on dim G, and that the full automorphism group Aut(X) of every projective variety X has the Jordan property.

Jordan Groups and Algebraic Surfaces

Transformation Groups, 2014

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of

Complex Tori and Automorphism Groups of certain P1P^1P1-bundles

2021

This paper is a continuation of our article [BZ20]. The notion of a poor complex compact manifold was introduced there and the group Aut(X) for a P1−bundle over such a manifold was proven to be very Jordan. We call a group G very Jordan if it contains a normal abelian subgroup G0 such that the orders of finite subgroups of the quotient G/G0 are bounded by a constant depending on G only. In this paper we provide examples of infinite families of poor manifolds of any complex dimension, namely simple tori of algebraic dimension zero. Then we consider a non-trivial holomorphic P1−bundle (X, p, Y ) over a non-uniruled complex compact Kähler manifold Y. We prove that Aut(X) is very Jordan provided some additional conditions on the set of sections of p are met. Applications to P-bundles over non-algebraic complex tori are given.

Jordan tori for a torsion free abelian group

Frontiers of Mathematics in China, 2014

We classify Jordan G-tori, where G is any torsion-free abelian group. Using the Zelmanov prime structure theorem, such a class divides into three types, namely, the Hermitian type, the Clifford type and the Albert type. We concretely describe Jordan G-tori of each type.

Jordan Properties of Automorphism Groups of Certain Open Algebraic Varieties

Transformation Groups, 2018

Let W be a quasiprojective variety over an algebraically closed field of characteristic zero. Assume that W is birational to a product of a smooth projective variety A and the projective line. We prove that if A contains no rational curves then the automorphism group G := Aut(W) of W is Jordan. That means that there is a positive integer J = J(W) such that every finite subgroup B of G contains a commutative subgroup A such that A is normal in B and the index [B : A] ≤ J .

Anti-tori in Square Complex Groups

Geometriae Dedicata, 2005

An anti-torus is a subgroup a, b in the fundamental group of a compact non-positively curved space X, acting in a specific way on the universal covering spaceX such that a and b do not have any commuting nontrivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups p,l originally studied by Mozes [Israel J. Math. 90(1-3) (1995), 253-294]. It turns out that anti-tori in p,l directly correspond to noncommuting pairs of Hamilton quaternions. Moreover, free anti-tori in p,l are related to free groups generated by two integer quaternions, and also to free subgroups of SO 3 (Q). As an application, we prove that the multiplicative group generated by the two quaternions 1 + 2i and 1 + 4k is not free.

Jordan groups, conic bundles and abelian varieties

Algebraic Geometry, 2017

A group G is called Jordan if there is a positive integer J = J G such that every finite subgroup B of G contains a commutative subgroup A ⊂ B such that A is normal in B and the index [B : A] is at most J (V. L. Popov). In this paper, we deal with Jordan properties of the groups Bir(X) of birational automorphisms of irreducible smooth projective varieties X over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov and C. Shramov) that Bir(X) is Jordan if X is non-uniruled. On the other hand, the second-named author proved that Bir(X) is not Jordan if X is birational to a product of the projective line P 1 and a positive-dimensional abelian variety. We prove that Bir(X) is Jordan if (the uniruled variety) X is a conic bundle over a non-uniruled variety Y but is not birational to Y × P 1. (Such a conic bundle exists if and only if dim(Y) 2.) When Y is an abelian surface, this Jordan property result gives an answer to a question of Prokhorov and Shramov.

Jordan groups and elliptic ruled surfaces

arXiv (Cornell University), 2014

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir

On two-generator subgroups of mapping torus groups

arXiv (Cornell University), 2024

We prove that if Gϕ = F, t|txt -1 = ϕ(x), x ∈ F is the mapping torus group of an injective endomorphism ϕ : F → F of a free group F (of possibly infinite rank), then every twogenerator subgroup H of Gϕ is either free or a sub-mapping torus. As an application we show that if ϕ ∈ Out(Fr) (where r ≥ 2) is a fully irreducible atoroidal automorphism then every two-generator subgroup of Gϕ is either free or has finite index in Gϕ.

Embeddings of rank-2 tori in algebraic groups

Journal of Pure and Applied Algebra, 2018

Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type A 2 , G 2 and F 4 defined over k, via their rank-2 k-tori. Simple, simply connected groups of type A 2 play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of A n type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type A 2 , G 2 and F 4 in terms of the mod-2 Galois cohomological invariants attached with these groups. We calculate the number of rank-2 k-unitary tori generating these algebraic groups (in fact exhibit such tori explicitly). The results in this paper and our earlier paper ([8]) show that the mod-2 invariants of groups of type G 2 , F 4 and A 2 are controlled by their k-subgroups of type A 1 and A 2 as well as the unitary k-tori embedded in them.

A Jordan-Hoelder Theorem for Differential Algebraic Groups

2010

We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a uniqueness result, prove several properties of almost simple groups and, in the ordinary differential case, classify almost simple linear differential algebraic

Rationality Problem for Generic Tori in Simple Groups

Journal of Algebra, 2000

We prove that except for several already known cases, the generic torus of a simple (adjoint or simply connected) group is not stably rational. This confirms a conjecture by Le Bruyn on generic norm tori.