Grammar development over the last decades has seen a shift away from large inventories of grammar rules to richer lexical structures. Many modern grammar theories are highly lexicalised. But simply listing lexical entries typically results in an undesirable amount of redundancy. Lexical inheritance hierarchies, on the other hand, make it possible to capture linguistic generalisations and thereby reduce redundancy. Inheritance hierarchies are usually constructed by hand but this is time-consuming and often impractical if a lexicon is very large. Constructing hierarchies automatically or semiautomatically facilitates a more systematic analysis of the lexical data. In addition, lexical data is often extracted automatically from corpora and this is likely to increase over the coming years. Therefore it makes sense to go a step further and automate the hierarchical organisation of lexical data too. Previous approaches to automatic lexical inheritance hierarchy construction tended to focus on minimality criteria, aiming for hierarchies that minimised one or more criteria such as the number of path-value pairs, the number of nodes or the number of inheritance links (Petersen 2001, Barg 1996a, and in a slightly different context: Light 1994). Aiming for minimality is motivated by the fact that the conciseness of inheritance hierarchies is a main reason for their use. However, I will argue that there are several problems with minimality-based approaches. First, minimality is not well defined in the context of lexical inheritance hierarchies as there is a tension between different minimality criteria. Second, minimality-based approaches tend to underestimate the importance of linguistic plausibility. While such approaches start with a definition of minimal redundancy and then try to prove that this leads to plausible hierarchies, the approach suggested here takes the opposite direction. It starts with a manually built hierarchy to which a supervised machine learning algorithm is applied with the aim of finding a set of formal criteria that can guide the construction of plausible hierarchies. Taking this direction means that it is more likely that the selected criteria do in fact lead to plausible hierarchies. Using a machine learning technique also has the advantage that the set of criteria can be much larger than in hand-crafted definitions. Consequently, one can define conciseness in very broad terms, taking into account interdependencies in the data as well as simple minimality criteria. This leads to a more fine-grained model of hierarchy quality. In practice, the method proposed here consists of two components: Galois lattices are used to define the search space as the set of all generalisations over the input lexicon. Maximum entropy models which have been trained on a manually built hierarchy are then applied to the lattice of the input lexicon to distinguish between plausible and implausible generalisations based on the formal criteria that were found in the training step. An inheritance hierarchy is then derived by pruning implausible generalisations. The hierarchy is automatically evaluated by matching it to a manually built hierarchy for the input lexicon. Automatically constructing lexical hierarchies is a hard task, partly because what is considered the best hierarchy for a lexicon is to some extent subjective. Supervised learning methods also suffer from a lack of suitable training data. Hence, a semi-automatic architecture may be best suited for the task. Therefore, the performance of the system has been tested using a semi-automatic as well as an automatic architecture and it has also been compared to the performance achieved by the pruning algorithm suggested by Petersen (2001). The findings show that the method proposed here is well suited for semi-automatic hierarchy construction.

<p dir="ltr">A group is said to be matricially (Frobenius) stable if every function from the group to unitary matrices that is "almost multiplicative" in the point operator (Frobenius) norm topology is "close" to a genuine unitary representation in the same topology. A result of Dadarlat shows that for a large class of groups, non-torsion even cohomology obstructs matricial stability. However, the proof doesn't generate explicit almost multiplicative maps that are far from genuine representations. In this paper, we compute explicit almost homomorphisms for all finitely generated groups with a non-torsion 2-cohomology class with a residually finite central extension. We use similar techniques to show that finitely generated nilpotent groups are Frobenius stable if and only if they are virtually cyclic, and that a finitely generated group with a non-torsion 2-cohomology class that can be written as a cup product of two 1-cohomology classes is not Frobenius stable.</p><p><br></p>

No description available.

No description available.

No description available.

No description available.

Extending an explicit result from Bridson–Conder–Reid, this work provides an algorithm for distinguishing finite quotients between cocompact triangle groups Δ ?and lattices Γ of constant curvature symmetric 2-spaces. Much of our attention will be on when these lattices are Fuchsian groups. We prove that it will suffice to take a finite quotient that is Abelian, dihedral, a subgroup of PSL(<i>n</i>,<b>F</b>_<i>q</i>) (for an odd prime power q), or an Abelian extension of one of these 3 groups. For the latter case, we will require and develop an approach for creating group extensions upon a shared finite quotient of Δ? and Γ which between them have differing degrees of smoothness. Furthermore, on the order of a finite quotient that distinguishes between ?Δ and Γ, we are able to establish an effective upperbound that is superexponential depending on the cone orders appearing in each group.<br>

<p>Whether the growth series of a group is a rational function is investigated in this paper.Parry showed certain torus bundle groups of even trace exhibits rational growth, and thisresult has been extended by the author, Turbo Ho and Mark Pengitore. In this paper, bothresults are combined into a single proof used in [1], and the result is pushed further into thenegative case not covered in earlier works</p>

<p dir="ltr">In this article, we establish results concerning the nilpotent representation theory of groups. In particular, we utilize a theorem of Stallings to provide a general method that constructs pairs of groups that have isomorphic universal nilpotent quotients. We then prove by counterexample that absolute Galois groups of number fields are not determined by their universal nilpotent quotients. We also show that this is the case for residually nilpotent Kleinian groups and in fact, there exist non-isomorphic pairs that have arbitrarily large nilpotent genus. We additionally provide examples of non-isomorphic curves whose geometric fundamental groups have isomorphic universal nilpotent quotients and the isomorphisms are compatible with the outer Galois actions. </p>