Abstract Commensurators of Groups Acting on Rooted Trees

We show that the abstract commensurator of a nearly level transitive weakly branch group H coincides with the relative commensurator of H in the homeomorphism group of the boundary of the tree on which H acts. It is also shown that the commensurator of an infinite group which is commensurable with its own nth direct power contains a Higman–Thompson group as a subgroup. Applying these results to the Grigorchuk 2-group G we show that the commensurator of G is a finitely presented infinite simple group.     

Topological simplicity,commensurator super-rigidity and nonlinearities of Kac-Moody groups (appendix by P. Bonvin)

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.     

Commensurability of geometric subgroups of mapping class groups

Let M be an orientable surface with punctures and/or boundary components. Paris and Rolfsen (J Reine Angew Math 521:47–83, 2000) studied “geometric subgroups” of the mapping class group of M, that is subgroups corresponding to inclusions of connected subsurfaces. In the present paper we extend this analysis to disconnected subsurfaces and to the nonorientable case. We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, following the work of Paris (Math Ann 322:301–315, 2002), we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.     

Commensurations of Out(F<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

Let Out(F _n ) denote the outer automorphism group of the free group F _n with n>3. We prove that for any finite index subgroup Γ<Out(F _n ), the group Aut(Γ) is isomorphic to the normalizer of Γ in Out(F _n ). We prove that Γ is co-Hopfian: every injective homomorphism Γ→Γ is surjective. Finally, we prove that the abstract commensurator Comm(Out(F _n )) is isomorphic to Out(F _n ).     

Automata and Square Complexes

We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method we give examples of free groups and of Kazhdan groups which are generated by the different states of one finite (bi-reversible) automaton. We also reprove the theorem of Macedońska, Nekrashevych, Sushchansky, on the connection between bi-reversible automata and the commensurator of a regular tree.     

Constructing irreducible representations of discrete groups

The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasi-regular representations are irreducible if and only if the corresponding subgroups are self-commensurizing. The purpose of this work is to describe general constructions of pairs of groups Γ₀ with Γ its own commensurator in Γ. These constructions are then applied to groups of isometries of hyperbolic spaces and to lattices in algebraic groups.     

Superrigidity for irreducible lattices and geometric splitting

We prove general superrigidity results for actions of irreducible lattices on CAT spaces, first in terms of the ideal boundary, and then for the intrinsic geometry (also for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Semicharacters of Groups

We define the notion of a semicharacter of a group G: A function from the group to ?*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is divisible by the order of the group. We prove that the conjecture holds for some important families of groups, including the Symmetric groups and the groups GL(2, q).     

Symplectic Automorphisms and the Picard Group of a K3 Surface

We consider the symplectic action of a finite group G on a K3 surface X. The Picard group of X has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then investigate the classification of symplectic actions by a fixed finite group, using moduli spaces of K3 surfaces with symplectic G-action.     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

The topology of a semisimple Lie group is essentially unique

We study locally compact group topologies on simple and semisimple Lie groups. We show that the Lie group topology on such a group S is very rigid: every “abstract” isomorphism between S and a locally compact and σ-compact group Γ is automatically a homeomorphism, provided that S is absolutely simple. If S is complex, then noncontinuous field automorphisms of the complex numbers have to be considered, but that is all. We obtain similar results for semisimple groups.     

On Noetherian Modules over Minimax Abelian Groups

We consider modules over minimax Abelian groups. We prove that if A is an Abelian minimax subgroup of the multiplicative group of a field k and if the subring K of the field k generated by the subgroup A is Noetherian, then the subgroup A is the direct product of a periodic group and a finitely generated group.     

Fusions of Character Tables II: p-Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. In a previous article, we gave examples of Camina pairs that fuse from abelian groups. In this article, we give more general examples of Camina triples that fuse from abelian groups. We use this result to give an example of a group which fuses from an abelian group, but which has a subgroup that does not. We also give an example of a powerful 2-group which does not fuse from an abelian group and of a regular 3-group which does not fuse from an abelian group.     

On reduction of p-groups

We define the notion of irreducibility of a pgroup and show how any pgroup G can be reduced to an irreducible group H. We show that G is realizable as the Galois group of a regular extension of Q(T) if H is. Finally, we give some sufficient conditions on the

number of generators of a pgroup and the structure of its Frattini subgroup for it to be reducible to the trivial group.     

On finite groups isospectral to simple symplectic and orthogonal groups

The spectrum of a finite group is the set of its element orders. Two groups are said to be isospectral if their spectra coincide. We deal with the class of finite groups isospectral to simple and orthogonal groups over a field of an arbitrary positive characteristic p. It is known that a group of this class has a unique nonabelian composition factor. We prove that this factor cannot be isomorphic to an alternating or sporadic group. We also consider the case where this factor is isomorphic to a group of Lie type over a field of the same characteristic p.     

Multivalued groups, their representations and Hopf algebras

This paper introduces the concept ofn-valued groups and studies their algebraic and topological properties. We explore a number of examples. An important class consists of those that we calln-coset groups; they arise as orbit spaces of groupsG modulo a group of automorphisms withn elements. However, there are many examples that do not arise from this construction. We see that the theory ofn-valued groups is distinct from that of groups with a given automorphism group. There are natural concepts of the action of ann-valued group on a space and of a representation in an algebra of operators. We introduce the (purely algebraic) notion of ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure. The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra. In general the group ring of ann-valued group is not ann-Hopf algebra but it is for ann-coset group constructed from an abelian group. Using the properties ofn-Hopf algebras we show that certain spaces do not admit the structure of ann-valued group and that certain commutativen-valued groups do not arise by applying then-coset construction to any commutative group.     

Automorphisms and Homotopies of Groupoids and Crossed Modules

This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid C as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of C, and use these to define a new notion of groupoid action. We then show that the automorphism group of a crossed module of groupoids C\mathcal{C}, in the case when the range groupoid is connected and the source group totally disconnected, may be determined from that of the crossed module of groups C_u\mathcal{C}_u formed by restricting to a single object u. Finally, we show that the group of homotopies of C\mathcal{C} may be determined once the group of regular derivations of C_u\mathcal{C}_u is known.     

Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2π, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic cone-manifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group [(PSL₂\mathbb R)\tilde]{\widetilde{{\it PSL}_2{\mathbb R}}} of the group of orientation-preserving isometries of \mathbb H²{{\mathbb H}^2} and Markoff moves arising from the action of the mapping class group on the character variety.