Arithmeticity vs. non-linearity for irreducible lattices

We establish an arithmeticity vs. non-linearity alternative for irreducible lattices in suitable product groups, such as for instance products of topologically simple groups. This applies notably to a (large class of) Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page

Frobenius groups of automorphisms and their fixed points

Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are shown to be close to the corresponding properties of $C_G(H)$. By using Clifford's theorem it is proved that the order $|G|$ is bounded in terms of $|H|$ and $|C_G(H)|$, the rank of $G$ is bounded in terms of $|H|$ and the rank of $C_G(H)$, and that $G$ is nilpotent if $C_G(H)$ is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of $G$ in the case of metacyclic $FH$. The exponent of $G$ is bounded in terms of $|FH|$ and the exponent of $C_G(H)$ by using Lazard's Lie algebra associated with the Jennings--Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of $G$ is bounded in terms of $|H|$ and the nilpotency class of $C_G(H)$ by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.Comment: 31 page

Quadratic equations over free groups are NP-complete

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete

Finitely presented wreath products and double coset decompositions

We characterize which permutational wreath products W^(X)\rtimes G are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X^2. On the one hand, this extends a result of G. Baumslag about standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.Comment: 21 pages; no figure. To appear in Geom. Dedicat

Free-algebra functors from a coalgebraic perspective

Given a set $\Sigma$ of equations, the free-algebra functor $F_{\Sigma}$ associates to each set $X$ of variables the free algebra $F_{\Sigma}(X)$ over $X$. Extending the notion of \emph{derivative} $\Sigma'$ for an arbitrary set $\Sigma$ of equations, originally defined by Dent, Kearnes, and Szendrei, we show that $F_\Sigma$ preserves preimages if and only if $\Sigma \vdash \Sigma'$, i.e. $\Sigma$ derives its derivative $\Sigma'$. If $F_\Sigma$ weakly preserves kernel pairs, then every equation $p(x,x,y)=q(x,y,y)$ gives rise to a term $s(x,y,z,u)$ such that $p(x,y,z)=s(x,y,z,z)$ and $q(x,y,z)=s(x,x,y,z)$. In this case n-permutable varieties must already be permutable, i.e. Mal'cev. Conversely, if $\Sigma$ defines a Mal'cev variety, then $F_\Sigma$ weakly preserves kernel pairs. As a tool, we prove that arbitrary $Set-$endofunctors $F$ weakly preserve kernel pairs if and only if they weakly preserve pullbacks of epis

A lattice in more than two Kac--Moody groups is arithmetic

Let $\Gamma$ be an irreducible lattice in a product of n infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic group over a local field and $\Gamma$ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n is at least 2: either $\Gamma$ is an S-arithmetic (hence linear) group, or it is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.Comment: Subsection 2.B was modified and an example was added ther

Elastic symmetries of defective crystals

I construct discrete and continuous crystal structures that are compatible with a given choice of dislocation density tensor, and (following Mal’cev) provide a canonical form for these discrete structures. The symmetries of the discrete structures extend uniquely to symmetries of corresponding continuous structures—I calculate these symmetries explicitly for a particular choice of dislocation density tensor and deduce corresponding constraints on energy functions which model defective crystals

A classification of the symmetries of uniform discrete defective crystals

Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The geometrical symmetries of these structures can be computed in terms of the changes of generators of the discrete subgroup which preserve the discrete set of points. Here a classification of the symmetries for the discrete subgroups of a particular class of three-dimensional solvable Lie group is presented. It is a fact that there are only three mathematically distinct types of Lie groups which model uniform defective crystals, and the calculations given here complete the discussion of the symmetries of the corresponding discrete structures. We show that those symmetries corresponding to automorphisms of the discrete subgroups extend uniquely to symmetries of the ambient Lie group and we regard these symmetries as (restrictions of) elastic deformations of the continuous defective crystal. Other symmetries of the discrete structures are classified as ‘inelastic’ symmetries

On the global existence of hairy black holes and solitons in anti-de Sitter Einstein-Yang-Mills theories with compact semisimple gauge groups

We investigate the existence of black hole and soliton solutions to four dimensional, anti-de Sitter (adS), Einstein-Yang-Mills theories with general semisimple connected and simply connected gauge groups, concentrating on the so-called 'regular case'. We here generalise results for the asymptotically flat case, and compare our system with similar results from the well researched adS su(N) system. We find the analysis differs from the asymptotically flat case in some important ways: the biggest difference is that for Λ < 0, solutions are much less constrained as r → ∞, making it possible to prove the existence of global solutions to the field equations in some neighbourhood of existing trivial solutions, and in the limit of |Λ| → ∞. In particular, we can identify non-trivial solutions where the gauge field functions have no zeroes, which in the su(N) case proved important to stability