Ordering groups and validity in lattice-ordered groups

An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order.     

The shuffle algebra on the factors of a word is free

The shuffle algebra generated by the factors of a given word is shown to be free, with transcendance degree equal to the dimension of a Lie algebra canonically associated to this word.     

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so‐called word‐based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k ‐tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so‐called graph‐based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph‐based distribution, while they are exponentially generic in the word‐based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Finitely generated invariants of Hopf algebras on free associative algebras

We show that the invariants of a free associative algebra of finite rank under a linear action of a finite-dimensional Hopf algebra generated by group-like and skew-primitive elements form a finitely generated algebra exactly when the action is scalar. This generalizes an analogous result for group actions by automorphisms obtained by Dicks and Formanek, and Kharchenko.     

Local symmetry rank bound for positive intermediate Ricci curvatures

Geometriae Dedicata - We study the speed of convergence to the asymptotic cone for a finitely generated nilpotent group endowed with a word metric. The first result on this theme is given by Burago...     

Regular principal factors in free objects in Rees-Sushkevich varieties

In a previous paper, the author showed how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges. In the main theorem, it is shown how these fundamental semigroups can be used to describe the regular principal factors of the free objects in certain Rees-Sushkevich varieties, namely, the varieties of semigroups that are generated by all completely 0-simple semigroups over groups in a variety of finite exponent. This approach is then used to solve the word problem for each of these varieties for which the corresponding group variety has solvable word problem.     

Occurrence problem for free solvable groups

We study the problem on the existence of an algorithm verifying whether systems of linear equations over a group ring of a free metabelian group are solvable. The occurrence problem for free solvable groups of derived length 3is proved undecidable. We give an example of a group with undecidable word problem which is finitely presented in a variety of solvable groups and is defined by the relations from the last commutator subgroup. Translated fromAlgebra i Logika, Vol. 34, No. 2, pp. 211-232, March-April, 1995.     

Generalizations of Fibonacci polynomials and free linear groups

The aim of this paper is to relate some generalized Fibonacci polynomials to the problem of knowing when a group (or semigroup) generated by some linear transformations is free.     

The degree of the special linear characters of a rank two free group

Given a word W in the free group on 2 letters F ₂, Horowitz showed that the special linear character of W is an integer polynomial in the 3 characters of the basic words of F ₂. Special linear characters are defined via the trace of their representations, and the polynomial character of an arbitrary W can be found by application of certain “trace relations”, which allow one to write the character of a complicated word as a sum of products of the characters of simpler words. Even in the n = 2 case, where the polynomial is uniquely defined, this procedure can be difficult and tedious. In this note, we use the structure of a free group word in F ₂ to compute the degree and the leading monomial of its SL(2,\mathbbC){SL(2,\mathbb{C})} -character without actually computing the full polynomial.     

Word maps in finite simple groups

Elements of the free group define interesting maps, known as word maps, on groups. It was previously observed by Lubotzky that every subset of a finite simple group that is closed under endomorphisms occurs as the image of some word map. We improve upon this result by showing that the word in question can be chosen to be in $$v(\mathbf F _n),$$ the verbal subgroup of the free group generated by the word v, provided that v is not a law on the finite simple group in question. In addition, we provide an example of a word w that witnesses the chirality of the Mathieu group $$M_{11}$$. The paper concludes by demonstrating that not every subset of a group closed under endomorphisms occurs as the image of a word map.     

Subalgebras of free algebras

We use non-commutative Jacobian matrix to get information on finitely generated subalgebras of a free Lie algebra. In particular, we show that the rank of such a subalgebra is equal to the left rank (i.e., to the maximal number of left independent rows) of the corresponding Jacobian matrix; this also yields an effective procedure for finding the rank of a finitely generated subalgebra.

     

An order topologyfor finitely generated free monoids

In this article a method is given for embedding a finitely generated free monoid as a dense subset of the unit interval. This gives an order topology for the monoid such that the submonoids generated by an important class of maximal codes occur as “thick” subsets. As an ordered topological space, the notion of thickness in a frec monoid can be interpreted in a number of ways. One such notion is that of density. In particular, subsets of a free monoid that fail to meet all two sided ideals (the thin sets, of which recognizable codes are an example) are shown (corollary 4.2) to be nowhere dense. Furthermore, it is shown (corollary 5.1) that a thin code is maximal if and only if the submonoid that it generates is dense on some interval. Thus thin codes that are maximal are precisely those that generate thick submonoids. Another notion of thickness is that of category. The embedding allows the free monoid to be viewed as a subspace of the unit interval. In theorem 5.6 it is shown that a thin code is maximal just in case the closure of the submonoid that it generates is second category in the unit interval. A mild connection with Lebesque measure is then made. In what follows, all free monoids are assumed to be generated by a finite set of at least two elements. IfA is such a set, thenA ^* denotes the free monoid generated byA. The setA is called an alphabet, the elements ofA ^* are called words, ande denotes the empty word inA ^*. Topological terminology and notation follows that of Kelley [2].     

Some algorithmic problems for relatively free groups

The paper contains the proof of the algorithmic unrecognizability for two properties of some relatively free soluble groups, as well as the proof of the lack in a certain sense of the recursive enumerability for one more property of such groups. It is also proved that the indecomposability in the class of soluble finitely based varieties of groups is unrecognizable. An example is given of a relatively free soluble group with solvable word problem and with unsolvable identity problem. A result is obtained involving truth-table degrees of the word problem for relatively free soluble groups. Bibliography: 22 titles.     

Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].     

On maximal subgroups of free idempotent generated semigroups

We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister-Schreier type rewriting.     

Subgroup properties of fully residually free groups

We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup.

     

Optimization over structured subsets of positive semidefinite matrices via column generation

We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.     

Unbounded generalizations of left Hilbert algebras,II

The first purpose of this paper is to study the classification of unbounded left Hilbert algebras. The second purpose is to investigate the unbounded left Hilbert algebras generated by positive linear functional on a ¹-algebra. The final purpose is to study the classification of positive linear functionals.     

Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition, we prove that the unique compact, orientable, second order minima of the weighted area under variations preserving the weighted volume and with free boundary in the boundary of the cone are intersections with the cone of round spheres centered at the vertex.