Conjugacy Classes Of Finite Subgroups Research Articles

Applications of the Fixed Point Theorem for group actions on buildings to algebraic groups over polynomial rings

We apply the Fixed Point Theorem for the actions of finite groups on Bruhat–Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable, assuming that the base field is of characteristic zero. First, we prove that for a reductive k-group G, every finite subgroup of G(k[t]) is conjugate to a subgroup of G(k). This, in particular, implies that if k is a finite extension of the p-adic field Qp, then the group G(k[t]) has finitely many conjugacy classes of finite subgroups, which is a well-known property for arithmetic groups. Second, we give a short proof of the theorem of Raghunathan–Ramanathan [26] about G-torsors over the affine line.

Finite subgroups of automorphisms of K3 surfaces

AbstractWe give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.

Finiteness of Klein actions and real structures on compact hyperkähler manifolds

One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperk\"ahler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperk\"ahler manifold contain only finitely many conjugacy classes of finite subgroups. We furthermore answer a question of Oguiso by showing that the automorphism group of a compact hyperk\"ahler manifold is finitely presented.

A trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity

Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel [6] combined with Mostow rigidity imply that there exists a finite group G=G(M) such that any finite subgroup of Homeo+(M) is isomorphic to a subgroup of G. Borel [6] asked if there exist M's with G(M) trivial and if the number of conjugacy classes of finite subgroups of Homeo+(M) is finite. We answer both questions:(1)For every finite group G there exist M's with G(M)=G, and(2)the number of maximal subgroups of Homeo+(M) can be either one, countably many or continuum and we determine (at least for dim⁡M≠4) when each case occurs. Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity (for dimM≠4) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.

Subgroups of odd order in the real plane Cremona group

In this paper we describe conjugacy classes of finite subgroups of odd order in the group of birational automorphisms of the real projective plane.

Dismantlability of weakly systolic complexes and applications

The main goal of this paper is proving the fixed point theorem for finite groups acting on weakly systolic complexes. As corollaries we obtain results concerning classifying spaces for the family of finite subgroups of weakly systolic groups and conjugacy classes of finite subgroups. As immediate consequences we get new results on systolic complexes and groups. The fixed point theorem is proved by using a graph-theoretical tool — dismantlability. In particular we show that 1 1 –skeleta of weakly systolic complexes, i.e., weakly bridged graphs, are dismantlable. On the way we show numerous characterizations of weakly bridged graphs and weakly systolic complexes.

Accessing the cohomology of discrete groups above their virtual cohomological dimension

We introduce a method to explicitly determine the Farrell–Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL2(O) for O the ring of integers in an imaginary quadratic number field, and to their finite index subgroups. We show that the Farrell–Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell–Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups have been determined in a thesis of Krämer, in terms of elementary number-theoretic information on O. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the electronically released appendix to this paper. Our new insights about their homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.

Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.

Fixed points of finite groups acting on generalised Thompson groups

We study centralisers of finite order automorphisms of the generalised Thompson groups Fn,∞ and conjugacy classes of finite subgroups in finite extensions of Fn,∞. In particular, we show that centralisers of finite automorphisms in Fn,∞ are either of type FP∞ or not finitely generated.

Morse theory and conjugacy classes of finite subgroups II

We construct a hyperbolic group containing a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.

Finite Subgroups in Algebras and Cohomology

We give a cohomological characterization of the set of conjugacy classes of finite subgroups of the projective multiplicative group of a finite dimensional algebra that become conjugate to a given group over some finite separable extension of the base field. We provide two applications.

Morse theory and conjugacy classes of finite subgroups

We construct a hyperbolic group containing a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.

AUTOMORPHISMS OF PRO-p GROUPS OF FINITE VIRTUAL COHOMOLOGICAL DIMENSION

AbstractLet G be a pro-p group of finite cohomological dimension and type FP ∞ and T be a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP ∞ (in particular it is finitely presented). Moreover, we prove that a pro-p group G of type FP ∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in PSL (2, R) ) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Cohomology and finite subgroups of profinite groups

We prove two theorems linking the cohomology of a pro-p group G with the conjugacy classes of its finite subgroups. The number of conjugacy classes of elementary abelian p-subgroups of G is finite if and only if the ring H*(G, Z/p) is finitely generated modulo nilpotent elements. If the ring H*(G,Z/p) is finitely generated, then the number of conjugacy classes of finite subgroups of G is finite.

Some groups of type VF

A group is of type VF if it contains a finite-index subgroup which has a finite classifying space. We construct groups of type VF in which the centralizers of some elements of finite order are not of type VF and groups of type VF containing infinitely many conjugacy classes of finite subgroups. From these examples it follows that a group G of type VF need not admit a finite-type or finite classifying space for proper actions (sometimes also called the universal proper G-space). We construct groups G for which the minimal dimension of a universal proper G-space is strictly greater than the virtual cohomological dimension of G. Each of our groups embeds in some general linear group over the rational integers. Applications to algebraic K-theory of group algebras and topological K-theory of group C*-algebras are also considered. The groups are constructed as finite extensions of Bestvina-Brady groups.

Finiteness properties for subgroups of GL $(n,\mathbb Z)$

Abstract. We construct finitely presented subgroups of GL $(n,{\mathbb Z})$ that have infinitely many conjugacy classes of finite subgroups. This answers a question of Grunewald and Platonov. We suggest a variation on their question.

Minimal index torsion-free subgroups of Kleinian groups

A Kleinian group Γ is a discrete subgroup of PSL(2, C), the full group of orientation-preserving isometries of 3-dimensional hyperbolic space. In the language of [T1] Q = H3/Γ is a hyperbolic 3-orbifold; that is a metric 3-orbifold in which all sectional curvatures are -1, and for which Γ is the orbifold fundamental group (see [T1] for further details). A Fuchsian group is a discrete subgroup of PSL(2, R) and as such acts discontinuously on the hyperbolic plane. We define Γ to be of finite co-volume (resp. co-area) if the volume (resp. area) of the quotient orbifold Q is finite. By Selberg’s Lemma, if Γ is a Kleinian (resp. Fuchsian) group of finite covolume (resp. co-area) it contains a torsion-free subgroup of finite index. By definition any torsion-free subgroup cannot contain any finite subgroups of Γ , so the index must be a multiple of the lowest common multiple of all orders of finite subgroups of Γ (see [CFJR] for a proof). Γ being of finite co-volume implies there are only finitely many conjugacy classes of finite subgroups. Thus, we make the following

On finite extensions of arithmetic groups

It is known that an arithmetic group has only finitely many conjugacy classes of finite subgroups. We generalize this result to groups which are finite extensions of arithmetic groups. We also prove that many (but not all) of these groups are again arithmetic groups. We construct a finitely generated subgroup of GL ( 4. ℤ ) which has infinitely many conjugacy classes of subgroups of order 4.

Conjugacy Classes of Finite Subgroups of Kleinian Groups

Conjugacy Classes of Finite Subgroups of Kleinian Groups