Geometric Group Theory

Non-Archimedean words have been introduced as a new type of infinite words which can be investigated through classical methods in combinatorics on words due to a length function. The length function, however, takes values in the additive group of polynomials ℤ[t] (and not, as traditionally, in ℕ), which yields various new properties. Non-Archimedean words allow to solve a number of interesting algorithmic problems in geometric and algorithmic group theory. There is also a connection to logic and the first-order theory in free groups (Tarski Problems). In the present paper we provide a general method to use infinite words over a discretely ordered abelian group as a tool to investigate certain group extensions for an arbitrary group G. The central object is a group E (A, G) which is defined in terms of a non-terminating, but confluent rewriting system. The groupG as well as some natural HNN-extensions of G embed into E (A, G) (and still "behave like" G), which makes it interesting to study its algorithmic properties. In order to show that every group G embeds into E (A, G) we combine methods from combinatorics on words, string rewriting and group theory.

In a context of special relativity under anisotropy, Burde recently presented in this journal a new group of coordinate transformations obtained using Lie group techniques from the principles of relativity and correspondence, which show conformal metric invariance. As a consequence, this author argued that conformal invariance is a necessary condition of any anisotropic transformation satisfying those principles, and rejected the adequacy of any transformation obtained from the Lorentz transformation by a coordinate change, e.g. resynchronization such as the [Formula: see text]-Lorentz transformation. This paper argues against these assertions, and presents, as a counterexample, a relativistic generalization of the Tangherlini transformation obtained by the same Lie group method, that satisfies the previous requirements and enjoys the metric invariance property.

In this paper, the conformal invariance and conserved quantities for higher-order holonomic systems are studied. Firstly, by establishing the differential equation of motion for the systems and introducing a one-parameter infinitesimal transformation group together with its infinitesimal generator vector, the determining equation of conformal invariance for the systems are provided, and the conformal factors expression are deduced. Secondly, the relation between conformal invariance and the Lie symmetry by the infinitesimal one-parameter point transformation group for the higher-order holonomic systems are deduced. Thirdly, the conserved quantities of the systems are derived using the structure equation satisfied by the gauge function. Lastly, an example of a higher-order holonomic mechanical system is discussed to illustrate these results.

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

In ?1 of this paper it is shown that a variety of conditions implying nice behavior for topological transformation groups are, in the presence of separability, equivalent. In ?2 the continuity properties of the stability subgroups are studied. The conditions of ?1 exclude the line acting on the torus in such a way that each orbit is dense. They exclude the integers acting on the circle by rotation through multiples of an irrational angle and they exclude the group of those sequences of zeros and ones which have all but a finite number of their terms equal to zero when this group acts on the space of all sequences of zeros and ones by coordinatewise addition (mod 2). As we shall see in the proof of Theorem 1, the latter transformation group is a prototype for all excluded transformation groups. This is analogous to the following fact in the theory of Rings of Operators: Every factor of type II, contains a hyperfinite factor of type II,. The conditions were suggested by [3, Theorem 1] and the proof of their equivalence is somewhat analogous to the proof of [3, Theorem 1]. However, the proof does not depend upon [3] nor upon the theory of C*-algebras.

Let X be a Riemann surface which admits a group H of automorphisms and assume that X is a branched cover of its orbit space under H.We derive necessary and sufficient conditions which determine when an involution of the orbit space lifts to an automorphism of X. We extend the well known result that if the orbit space is hyperelliptic and H is abelian and fixed point free, then the hyperelliptic involution lifts to an automorphism of X; we prove that if no Weierstrass points of the orbit space are ramified in X, and if the ramification indices of all non-Weierstrass points are relatively prime in pairs, then the hyperelliptic involution lifts for any abelian group H. It is possible to use our methods to obtain conditions for lifting automorphisms of arbitrary order and we give the conditions for lifting automorphisms of order three.

A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d).

Chapter 1 Galois Theory: 1.1 Action of a Solvable Group and Representability by Radicals.- 1.2 Fixed Points under an Action of a Finite Group and Its Subgroups.- 1.3 Field Automorphisms and Relations between Elements in a Field.- 1.4 Action of a k-Solvable Group and Representability by k-Radicals.- 1.5 Galois Equations.- 1.6 Automorphisms Connected with a Galois Equation.- 1.7 The Fundamental Theorem of Galois Theory.- 1.8 A Criterion for Solvability of Equations by Radicals.- 1.9 A Criterion for Solvability of Equations by k-Radicals.- 1.10 Unsolvability of Complicated Equations by Solving Simpler Equations.- 1.11 Finite Fields.- Chapter 2 Coverings: 2.1 Coverings over Topological Spaces.- 2.2 Completion of Finite Coverings over Punctured Riemann Surfaces.- Chapter 3 Ramified Coverings and Galois Theory: 3.1 Finite Ramified Coverings and Algebraic Extensions of Fields of Meromorphic Functions.- 3.2 Geometry of Galois Theory for Extensions of a Field of Meromorphic Functions.- References.- Index

Given a Riemann surface S, there exists a finitely generated Fuchsian group G of the first kind acting on the upper half plane U, such that S≅U/G. This isomorphism makes it possible to use Fuchsian group methods to prove theorems about Riemann surfaces. In this note we give a proof of the Serre duality theorem by Fuchsian group methods which is technically simpler than proofs depending on sheaf theoretic methods.

We begin by considering two familiar ways of constructing Riemann surfaces. First, we take a power series P converging on some disc D0 with centre Z0, and expand P about some point z1 in Do other than z0. In general, P will converge in a disc D1 extending beyond D0? and if we continue this process indefinitely we obtain a maximal Riemann surface on which the analytic continuation of P is defined. Of course, if we return to a region where P is already defined, but with different values, we create a new ‘sheet’ of the surface; thus we are led to the notion of a Riemann surface constructed from a given power series: this is the Weierstrass approach. A more modern approach is simply to define a Riemann surface as a complex analytic manifold but either way, there is the problem of showing that these two definitions are equivalent. It is easy enough to see that a Riemann surface obtained by analytic continuation is an analytic manifold, so we must focus our efforts on showing that every analytic manifold supports an analytic function. One solution to this problem lies in showing first that every such manifold arises as the quotient by a group action, and second, that we can construct functions invariant under this group action. As a by-product of a study of these groups we obtain important and very detailed quantitative information about the geometric nature of the general Riemann surface.

The polymer quantization of cosmological backgrounds provides an alternative path to the original Wheeler-de Witt (WdW) quantum cosmology, based on a different representation the commutation relations of the canonical variables. This polymer representation allows to capture the lattice like structure of the quantum geometry and leads to a radically different quantum cosmology compared to the WdW construction. This new quantization scheme has attracted considerable attention due to the singularity resolution it allows in a wide class of symmetry reduced gravitational systems, in particular where the WdW scheme fails. However, as any canonical quantization scheme, ambiguities in the construction of the quantum theory, being regularization or factor-ordering ones, can drastically modify the resulting quantum dynamics. In this work, we propose a new criteria to restrict the quantization ambiguities in the simplest model of polymer quantum cosmology, for homogeneous and isotropic General Relativity minimally coupled to a massless scalar field. This new criteria is based on an underlying SL(2,ℝ) structure present in the phase space of this simple cosmological model. By preserving the symmetry of this cosmological system under this 1d conformal group, we derive a new regularization of the phase space. We perform both its polymer quantization and a quantization scheme directly providing a representation of the SL(2,ℝ) group action. The resulting quantum cosmology can be viewed as a lattice-like quantum mechanics with an SL(2,ℝ) invariance. This provides a new version of Loop Quantum Cosmology consistent with the conformal symmetry. This alternative construction opens new directions, among which a possible mapping with the conformal quantum mechanics as well as with recent matrix or tensor models constructions for quantum cosmological space-time.

This paper is concerned with the topological structure of solvmanifolds i.e. spaces on which solvable Lie groups of transformations act transitively. The special case of nilmanifolds in which the group of transformations is nilpotent has been studied by Malcev [5]. Examples of two dimensional solvmanifolds are the torus, Klein bottle, cylinder, mobius strip. The central results are THEOREM A. Two compact solvmanifolds having the same fundamental group are isomorphic. THEOREM B. Any solvmanifold is covered a finite number of times by the direct product of a compact solvmanifold and a euclidean space; moreover, the covering group is abelian. THEOREM C. If S is an algebraically subgroup (cf. ?9 for definition) of the solvable Lie group G, then G/S is homeomorphic to the direct product of a compact solvmanifold and a euclidean space. Theorem C applies for example, if S is a closed subgroup which is generated by its connected identity-component together with elements s1, s2, -such that all the eigenvalues of Adsi are positive (i = 1, 2, * ). The assertion in Theorem A, applied to nilmanifolds, was proved by Malcev [5]. The situation in that case is simplified by virtue of the fact that two nilpotent groups which operate transitively and effectively on spaces with same fundamental group are isomorphic-a fact which is not true in the case of general solvmanifolds. Accordingly, there is a natural homeomorphism between nilmanifolds with isomorphic fundamental groups but there is no natural homeomorphism in the case of solvmanifolds. In the absence of uniqueness, therefore, proof of the existence of the homeomorphisin in Theorem A is rather elusive. After a number of algebraic maneuvers, proof of Theorem A is reduced to comparing two fibre bundles over homeomorphic toroids-the fibres being the orbits of transformation groups which are isomorphic as abstract topological groups. The main topological lemma asserts the existence of an equivariant homeomorphism between the two bundles. The major algebraic fact which underlies our method is the theorem of ?5 on uniform subgroups of solvable groups. In Theorem B, the covering need not be univalent-as is illustrated by the mobius band. The sufficiency condition in Theorem C yields as a special case a result of Chevalley: the factor space of a solvable Lie group by a connected subgroup is a product of circles and straight lines. Theorem A yields more generally

The purpose of this paper is to give a self-contained exposition of the Atiyah–Bott picture [AB83] for the Yang–Mills equation over Riemann surfaces with an emphasis on the analogy to finite dimensional geometric invariant theory. The main motivation is to provide a careful study of the semistable and unstable orbits: This includes the analogue of the Ness uniqueness theorem for Yang–Mills connections, the Kempf–Ness theorem, the Hilbert–Mumford criterion and a new proof of the moment-weight inequality following an approach outlined by Donaldson [Don4]. A central ingredient in our discussion is the Yang–Mills ow for which we assume longtime existence and convergence (see [Rad]).

In this paper we introduce the new concept of the conformal invariance and the conserved quantities for Appell systems under second-class Mei symmetry. The one-parameter infinitesimal transformation group and infinitesimal transformation vector of generator are described in detail. The conformal factor in the determining equations under second-class Mei symmetry is found. The relationship between Appell system's conformal invariance and Mei symmetry are discussed. And Appell system's conformal invariance under second-class Mei symmetry may lead to corresponding Hojman conserved quantities when the conformal invariance satisfies some conditions. Lastly, an example is provided to illustrate the application of the result.

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups.