32 results found
Evans DM, Ghadernezhad Z, Tent K, 2016, Simplicity of the automorphism groups of some Hrushovski constructions, ANNALS OF PURE AND APPLIED LOGIC, Vol: 167, Pages: 22-48, ISSN: 0168-0072
Evans DM, Tsankov T, 2016, Free actions of free groups on countable structures and property (T), FUNDAMENTA MATHEMATICAE, Vol: 232, Pages: 49-63, ISSN: 0016-2736
Amato D, Evans DM, 2015, Infinite primitive and distance transitive directed graphs of finite out-valency, Journal of Combinatorial Theory Series B, Vol: 114, Pages: 33-50, ISSN: 1096-0902
We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.
Amato D, Evans DM, Truss JK, 2012, Classification of some countable descendant-homogeneous digraphs, Discrete Mathematics, Vol: 312, Pages: 911-919, ISSN: 0012-365X
For finite q, we classify the countable, descendant-homogeneous digraphs in which the descendant set of any vertex is a q-valent tree. We also give conditions on a rooted digraph Γ which allows us to construct a countable descendant-homogeneous digraph in which the descendant set of any vertex is isomorphic to Γ. © 2011 Elsevier B.V. All rights reserved.
Evans DM, Ferreira MS, 2012, The geometry of hrushovski constructions, ii. the strongly minimal case, Journal of Symbolic Logic, Vol: 77, Pages: 337-349, ISSN: 0022-4812
We investigate the isomorphism types of combinatorial geometries arising from Hrushovski's flat strongly minimal structures and answer some questions from Hrushovski's original paper. © 2012, Association for Symbolic Logic.
Evans DM, Ferreira MS, 2011, The geometry of Hrushovski constructions, I: The uncollapsed case, Annals of Pure and Applied Logic, Vol: 162, Pages: 474-488, ISSN: 0168-0072
An intermediate stage in Hrushovski's construction of flat strongly minimal structures in a relational language L produces ω-stable structures of rank ω. We analyze the pregeometries given by forking on the regular type of rank ω in these structures. We show that varying L can affect the (local) isomorphism type of the pregeometry, but not its finite subpregeometries. A sequel will compare these to the pregeometries of the strongly minimal structures. © 2011 Elsevier B.V.
Evans DM, Pastori E, 2011, Second cohomology groups and finite covers of infinite symmetric groups, Journal of Algebra, Vol: 330, Pages: 221-233, ISSN: 0021-8693
For Ω an infinite set, k≥2 and W the set of k-sets from Ω, there is a natural closed permutation group Γk which is a non-split extension 0→Z2W→Γk→Sym(Ω)→1. We classify the closed subgroups of Γk which project onto Sym(Ω). The question arises in model theory as a problem about finite covers, but here we formulate and solve it in algebraic terms. © 2011 Elsevier Inc.
Emms J, Evans DM, 2009, Constructing continuum many countable, primitive, unbalanced digraphs, Discrete Mathematics, Vol: 309, Pages: 4475-4480, ISSN: 0012-365X
We construct continuum many non-isomorphic countable digraphs which are highly arc transitive, have finite out-valency and infinite in-valency, and whose automorphism groups are primitive. © 2009 Elsevier B.V. All rights reserved.
Evans DM, Wong MWHO, 2009, Some remarks on generic structures, Journal of Symbolic Logic, Vol: 74, Pages: 1143-1154, ISSN: 0022-4812
We show that the N 0 -categorical structures produced by Hrushovski's predimension construction with a control function fit neatly into Shelah's SOP n hierarchy: if they are not simple, then they have SOP 3 and NSOP 4 . We also show that structures produced without using a control function can be undecidable and have SOP. © 2009, Association for Symbolic Logic.
Evans DM, 2008, Expansions of fields by angular functions, Journal of the Institute of Mathematics of Jussieu, Vol: 7, Pages: 735-750, ISSN: 1474-7480
The notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two 'counterexamples' from model theory: the non-classical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function. © 2008 Cambridge University Press.
Evans DM, Hewitt PR, 2006, Continuous cohomology of permutation groups on profinite modules, Communications in Algebra, Vol: 34, Pages: 1251-1264, ISSN: 0092-7872
Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups on profinite modules, see Ahlbrandt and Ziegler (1991), Evans (1997b), Evans et al. (1997), and Hodges and Pillay (1994), for example. We expand the module category in order to widen the cohomological toolkit. For an important class of groups we use these tools to establish criteria for finiteness of cohomology.
Evans DM, 2005, Tivial stable structures with non-trivial reducts, Journal of the London Mathematical Society, Vol: 72, Pages: 351-363, ISSN: 0024-6107
A new viewpoint is offered on some of the generic structures constructed using Hrushovski's predimensions and it is shown that they are natural reducts of quite straightforward trivial, one-based stable structures. © 2005 London Mathematical Society.
Evans DM, 2004, Block transitive Steiner systems with more than one point orbit, Journal of Combinatorial Designs, Vol: 12, Pages: 459-465, ISSN: 1063-8539
For all 'reasonable' finite t, k, and s, we construct a t-(script N sign 0 , k, 1) design and a group of automorphisms which is transitive on blocks and has s orbits on points. In particular, there is a 2-(script N sign 0 , 4, 1) design with a block-transitive group of automorphisms having two point orbits. This answers a question of P. J. Cameron and C. E. Praeger. The construction is presented in a purely combinatorial way, but is a by-product of a new way of looking at a model-theoretic construction of E. Hrushovski. © 2004 Wiley Periodicals, Inc.
Evans DM, 2003, Ample dividing, Journal of Symbolic Logic, Vol: 68, Pages: 1385-1402, ISSN: 0022-4812
We construct a stable one-based, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory which is n-ample for all natural numbers n, and does not interpret an infinite group.
Evans DM, 2002, א0-categorical structures with a predimension, Annals of Pure and Applied Logic, Vol: 116, Pages: 157-186, ISSN: 0168-0072
We give an axiomatic framework for the non-modular simple א 0 -categorical structures constructed by Hrushovski. This allows us to verify some of their properties (weak elimination of imaginaries, simplicity, the nature of forking, CM-triviality and stable forking) in a uniform way, and to show that these properties are preserved by iterations of the construction. © 2002 Elsevier Science B.V. All rights reserved.
Evans DM, Pantano ME, 2002, א<inf>0</inf>-categorical structures with arbitrarily fast growth of algebraic closure, Journal of Symbolic Logic, Vol: 67, Pages: 897-909, ISSN: 0022-4812
Evans DM, Rashwan OA, 2002, Bounds in the theory of finite covers, Journal of Algebra, Vol: 250, Pages: 757-777, ISSN: 0021-8693
We give an upper bound for the number of conjugacy classes of closed subgroups of the full wreath product FWr W Sym(Ω) which project onto Sym(Ω). Here, Ω is infinite, W is the set of n-tuples of distinct elements from Ω (for some finite n), F is a finite nilpotent group, and the topology on the wreath product is that of pointwise convergence in its imprimitive permutation action. The result addresses a problem which arises in a natural model-theoretic context about classifying certain types of finite covers. © 2002 Elsevier Science (USA).
Brookes CJB, Evans DM, 2001, Augmentation modules for affine groups, Mathematical Proceedings of the Cambridge Philosophical Society, Vol: 130, Pages: 287-294, ISSN: 0305-0041
We prove that if k, F are fields of different characteristics then the permutation module kF n for the affine group AGL(n, F) has simple augmentation submodule. This proves a conjecture of A. R. Camina. © 2001 Cambridge Philosophical Society.
Evans DM, 2001, Suborbits in infinite primitive permutation groups, Bulletin of the London Mathematical Society, Vol: 33, Pages: 583-590, ISSN: 0024-6093
For every infinite cardinal κ. we construct a primitive permutation group which has a finite suborbit paired with a suborbit of size κ. This answers a question of Peter M. Neumann.
Evans DM, Wagner FO, 2000, Supersimple ω-categorical groups and theories, Journal of Symbolic Logic, Vol: 65, Pages: 767-776, ISSN: 0022-4812
An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl(Ø)-definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU-rank.
Evans DM, 1997, An infinite highly arc-transitive digraph, European Journal of Combinatorics, Vol: 18, Pages: 281-286, ISSN: 0195-6698
Evans DM, 1997, Computation of first cohomology groups of finite covers, Journal of Algebra, Vol: 193, Pages: 214-238, ISSN: 0021-8693
We give several applications of standard methods of group cohomology to some problems arising in model theory concerning finite covers. We prove a conjecture of the author that for G-finite, א 0 -categorical structures the kernels of minimal superlinked finite covers have bounded rank. We show that the cohomology groups associated to finite covers of certain structures (amongst them, the primitive, countable, totally categorical structures) have to be finite. From this we deduce that the finite covers of these structures are determined up to finitely many possibilities by their kernels. © 1997 Academic Press.
Evans DM, 1997, Finite covers with finite kernels, Annals of Pure and Applied Logic, Vol: 88, Pages: 109-147, ISSN: 0168-0072
We are concerned with the following problem. Suppose Γ and Σ are closed permutation groups on infinite sets C and W and ρ : Γ → Σ is a non-split, continuous epimorphism with finite kernel. Describe (for fixed Σ) the possibilities for ρ. Here, we consider the case where ρ arises from a finite cover π:C→W. We give reasonably general conditions on the permutation structure 〈W;Σ〉 which allow us to prove that these covers arise in two possible ways. The first way, reminiscent of covers of topological spaces, is as a covering of some Σ-invariant digraph on W. The second construction is less easy to describe, but produces the most familiar of these types of covers: a vector space covering its projective space.
Evans DM, 1996, Splitting of finite covers of א<inf>0</inf>-categorical structures, Journal of the London Mathematical Society, Vol: 54, Pages: 210-226, ISSN: 0024-6107
Suppose that W is a countable א 0 -categorical structure. We investigate the question as to whether every finite cover of W splits, that is, has an expansion which is a trivial finite cover of W. We show that for most primitive structures W which are homogeneous for a single binary relation (homogeneous graphs, partial orderings, the Henson digraphs, ...) any finite cover splits. However, in contrast to this, we show that there are non-split covers (with finite kernels) of the countable, universal, homogeneous local order.
Evans DM, Hrushovski E, 1995, The automorphism group of the combinatorial geometry of an algebraically closed field, Journal of the London Mathematical Society, Vol: 52, Pages: 209-225, ISSN: 0024-6107
Evans DM, Hrushovski E, 1993, On the automorphism groups of finite covers, Annals of Pure and Applied Logic, Vol: 62, Pages: 83-112, ISSN: 0168-0072
We are concerned with identifying by how much a finite cover of an א 0 -categorical structure differs from a sequence of free covers. The main results show that (in the best circumstances) this is measured by automorphism groups which are nilpotent-by-abelian. In the language of covers, these results say that every finite (regular) cover can be decomposed naturally into linked, superlinked and free covers. The superlinked covers arise from covers over a different base, and to describe this properly we introduce the notion of a quasi-cover. These results generalise results of the second author obtained in the case where the base of the cover is a grassmannian of a disintegrated set. They also give a complete proof of a statement of the second author extending this case to the case of a grassmannian of a modular set. To do this, we need to analyse the possible superlinked covers of such a set. We also give a combinatorial condition on the base of a cover which guarantees various chain conditions on finite covers over this base, and introduce a pregeometry which is useful in the analysis of finite covers with simple fibre groups. © 1993.
Evans DM, Siemons J, 1993, On the number of orbits of a group in two permutation actions, Archiv der Mathematik, Vol: 60, Pages: 420-424, ISSN: 0003-889X
Camina AR, Evans DM, 1991, Some infinite permutation modules, Quarterly Journal of Mathematics, Vol: 42, Pages: 15-26, ISSN: 0033-5606
Evans DM, 1991, The small index property for infinite dimensional classical groups, Journal of Algebra, Vol: 136, Pages: 248-264, ISSN: 0021-8693
We show that the countable dimensional analogues of the finite classical groups have the strong small index property. That is, if G is one of these groups and V is its natural module, and if H is a subgroup of G of index less than 2 ω then there exists a finite dimensional subspace X of V such that H is sandwiched between the pointwise and setwise stabilisers in G of X. © 1991.
Evans DM, Hodges W, Hodkinson IM, 1991, Automorphisms of Bounded Abelian Groups, Forum Mathematicum, Vol: 3, Pages: 523-542, ISSN: 0933-7741
We show that if A is a countable abelian group of finite exponent, and H is a subgroup of index less than 2 ω in the automorphism group Aut(A) of A, then H contains the pointwise stabiliser of some finite set of elements of A. (Thus A has the “small index property.”) We consider groups B which are direct sums of isomorphic cyclic p-groups Z (p m ), together with subgroups A which are the corresponding sums of p-groups Z (/p) for some t < m. For these pairs of groups we determine exactly when it is true that each automorphism of A extends to an automorphism of B, so that the extensions form an embedding of Aut(A) in Aut(B). Finally we discuss several connections between these results and questions in set theory and model theory. © de Gruyter 1991
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