International Journal of Algebra and Computation

A quasivariety is a universal Horn class of algebraic structures containing the trivial structure. The set of all subquasivarieties of a quasivariety forms a complete lattice under inclusion. A lattice isomorphic to for some quasivariety is called a lattice of quasivarieties or a quasivariety lattice. The Birkhoff–Maltsev Problem asks which lattices are isomorphic to lattices of quasivarieties. A lattice L is called unreasonable if the set of all finite sublattices of L is not computable, that is, there is no algorithm for deciding whether a finite lattice is a sublattice of L. The main result of this paper states that for any signature σ containing at least one non-constant operation, there is a quasivariety of signature σ such that the quasivariety lattice is unreasonable. Moreover, there are uncountable unreasonable lattices of quasivarieties. We also present some corollaries of the main result.

In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

Let K be a field and let UT_n(K) and T_n(K) denote the groups of all unitriangular and triangular matrices over field K, respectively. In the paper, the lattices of verbal subgroups of the above groups are characterized. Consequently, the equalities between certain verbal subgroups and their verbal width are determined. The considerations bring a series of verbal subgroups with exactly known finite width equal to 2. An analogous characterization and results for the groups of infinitely dimensional triangular and unitriangular matrices are established in the last part of the paper.

The semirings admitting maximal factorizations of any finite dimension are called MF-semirings. We first show that a commutative semiring K is an MF-semiring if and only if K admits a maximal factorization of dimension n ≥ 2, and if and only if K is a multiplicatively cancellative semiring satisfying the g.c.d. condition. And then, by using above result, we prove that a weighted finite automaton over a commutative idempotent MF-semiring has a determination if has the victory property and twins property. Also, some special cases are considered.

We say a subset Σ ⊆ F_N of the free group of rank N is spectrally rigid if whenever T₁, T₂ ∈ cv_N are ℝ-trees in (unprojectivized) outer space for which ‖σ‖_T1 = ‖σ‖_T2 for every σ ∈ Σ, then T₁ = T₂ in cv_N. The general theory of (non-abelian) actions of groups on ℝ-trees establishes that T ∈ cv_N is uniquely determined by its translation length function ‖⋅‖_T : F_N → ℝ, and consequently that F_N itself is spectrally rigid. Results of Smillie and Vogtmann, and of Cohen, Lustig and Steiner establish that no finite Σ is spectrally rigid. Capitalizing on their constructions, we prove that for any Φ ∈ Aut(F_N) and g ∈ F_N, the set Σ = {Φⁿ(g)}_n∈ℤ is not spectrally rigid.

This work lies across three areas of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link led to the calculation of some Kronecker coefficients by computing constant terms and conversely the computations of certain constant terms by computing Kronecker coefficients by symmetric function methods. This led to results as well as methods for solving numerical problems in each of these separate areas.

Let UT_n (K) be a unitriangular group over a field K. We prove that the group of all class preserving automorphisms of UT_n (K) is equal to Inn(UT_n (K)) if and only if K is a prime field. Let , where γ₃ (UT_n(𝔽_p^m)) denotes the third term of the lower central series of UT_n(𝔽_p^m). We calculate the group of all class preserving automorphisms and class preserving outer automorphisms of .

Each factor semigroup of a free restriction (ample) semigroup over a congruence contained in the least cancellative congruence is proved to be embeddable into a W-product of a semilattice by a monoid. Consequently, it is established that each restriction semigroup has a proper (ample) cover embeddable into such a W-product.

We study the Hilbert scheme of points on a cusp curve, mostly the punctual Hilbert scheme which parameterizes m points supported at a cusp. We show that the reduced punctual scheme is isomorphic to ℙ¹ and the Hilbert scheme has one singularity along .

We prove that for any finite index subgroup Γ in SL_n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL_n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γ_n(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than , where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.