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David M. Evans

Publications and source records attributed to David M. Evans.

17 recordsLinked to original sources
arXiv2026

Permutation modules for Ramsey structures

Suppose $R$ is a commutative ring and $G$ is a group acting on a set $W$. We consider the $RG$-module $RW$ in the case where $G$ is the automorphism group of an $ω$-categorical structure $M$ and $W$ is, for example, $M^n$ (for $n \in \mathbb{N}$). We develop methods which may provide information about two questions in the case where $R$ is a field $F$: whether $FW$ has a.c.c. on submodules; and in the case where $M$ is finitely homogeneous, whether $FW$ is of finite composition length. In the case where $M$ is a Ramsey structure and so $G$ is extremely amenable, we give a simple `decision procedure' for membership in a submodule of $RW$ specified by a given generating set. If $F$ is a field, we show that there is a duality between submodules of $FW$ and the topological $FG$-module of definable functions from $W$ to $F$.

math.GR↗
arXiv2022

Higher amalgamation properties in measured structures

Using an infinitary version of the Hypergraph Removal Lemma due to Towsner, we prove a model-theoretic higher amalgamation result. In particular, we obtain an independent amalgamation property which holds in structures which are measurable in the sense of Macpherson and Steinhorn, but which is not generally true in structures which are supersimple of finite SU-rank. We use this to show that some of Hrushovski's non-locally-modular, supersimple $ω$-categorical structures are not MS-measurable.

math.LO↗
arXiv2021

Simplicity of the automorphism groups of generalised metric spaces

Tent and Ziegler proved that the automorphism group of the Urysohn sphere is simple and that the automorphism group of the Urysohn space is simple modulo bounded automorphisms. A key component of their proof is the definition of a stationary independence relation (SIR). In this paper we prove that the existence of a SIR satisfying some extra axioms is enough to prove simplicity of the automorphism group of a countable structure. The extra axioms are chosen with applications in mind, namely homogeneous structures which admit a "metric-like amalgamation", for example all primitive 3-constrained metrically homogeneous graphs of finite diameter from Cherlin's list.

math.GR↗
arXiv2020

Ramsey properties and extending partial automorphisms for classes of finite structures

We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Nešetřil-Rödl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results solve several conjectures and provide an easy Ramseyness test for many classes of structures.

math.CO↗
arXiv2019

Permutation monoids and MB-homogeneity for graphs and relational structures

In this paper, we investigate the connection between infinite permutation monoids and bimorphism monoids of first-order structures. Taking our lead from the study of automorphism groups of structures as infinite permutation groups and the more recent developments in the field of homomorphism-homogeneous structures, we establish a series of results that underline this connection. Of particular interest is the idea of MB-homogeneity; a relational structure $\mathcal{M}$ is MB-homogeneous if every monomorphism between finite substructures of $\mathcal{M}$ extends to a bimorphism of $\mathcal{M}$. The results in question include a characterisation of closed permutation monoids, a Fraïssé-like theorem for MB-homogeneous structures, and the construction of $2^{\aleph_0}$ pairwise non-isomorphic countable MB-homogeneous graphs. We prove that any finite group arises as the automorphism group of some MB-homogeneous graph and use this to construct oligomorphic permutation monoids with any given finite group of units. We also consider MB-homogeneity for various well-known examples of homogeneous structures and in particular give a complete classification of countable homogeneous undirected graphs that are also MB-homogeneous.

math.GR↗
arXiv2019

Automorphism groups and Ramsey properties of sparse graphs

We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an $ω$-categorical sparse graph has no $ω$-categorical expansion with extremely amenable automorphism group.

math.GR↗
arXiv2018

Higher amalgamation properties in stable theories

For a complete, stable theory $T$ we construct, in a reasonably canonical way, a related stable theory $T^*$ which has higher independent amalgamation properties over the algebraic closure of the empty-set. The theory $T^*$ is an algebraic cover of $T$ and we give an explicit description of the finite covers involved in the construction of $T^*$ from $T$. This follows an approach of E. Hrushovski. If $T$ is almost strongly minimal with a $0$-definable strongly minimal set, then we show that $T^*$ has higher amalgamation over any algebraically closed subset.

math.LO↗
arXiv2017

Algorithms determining finite simple images of finitely presented groups

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, given any fixed untwisted Lie type $X$ there is an algorithm that determines whether or not an arbitrary finitely presented group has infinitely many simple images isomorphic to $X(q)$ for some $q$, and if there are finitely many, the algorithm determines them.

math.GR↗
arXiv2017

Finite homogeneous geometries

This paper reproduces the text of a part of the Author's DPhil thesis. It gives a proof of the classification of non-trivial, finite homogeneous geometries of sufficiently high dimension which does not depend on the classification of the finite simple groups.

math.GR↗
arXiv2015

Simplicity of the automorphism groups of some Hrushovski constructions

We show that the automorphism groups of certain countable structures obtained using the Hrushovski amalgamation method are simple groups. The structures we consider are the 'uncollapsed' structures of infinite Morley rank obtained by the ab initio construction and the (unstable) omega-categorical pseudoplanes. The simplicity of the automorphism groups of these follows from results which generalize work of Lascar and of Tent and Ziegler.

math.LO↗
arXiv2015

Free actions of free groups on countable structures and property (T)

We show that if $G$ is a non-archimedean, Roelcke precompact, Polish group, then $G$ has Kazhdan's property (T). Moreover, if $G$ has a smallest open subgroup of finite index, then $G$ has a finite Kazhdan set. Examples of such $G$ include automorphism groups of countable $ω$-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation groups. The latter allows the construction of a non-abelian free subgroup of $G$ acting freely in all infinite transitive permutation representations of $G$.

math.GR↗
arXiv2014

Infinite primitive and distance transitive directed graphs of finite out-valency

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.

math.CO↗
arXiv2011

Matroid Theory and Hrushovski's Predimension Construction

We give an exposition of some results from matroid theory which characterise the finite pregeometries arising from Hrushovski's predimension construction. As a corollary, we observe that a finite pregeometry which satisfies Hrushovski's flatness condition arises from a predimension.

math.LO↗
arXiv2011

Classification of some countable descendant-homogeneous digraphs

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 G which allow us to construct a countable descendant-homogeneous digraph in which the descendant set of any vertex is isomorphic to G.

math.CO↗
arXiv2011

Second cohomology groups and finite covers

For D an infinite set, k>1 and W the set of k-sets from D, there is a natural closed permutation group G_k which is a non-split extension of \mathbb{Z}_2^W by \Sym(D). We classify the closed subgroups of G_k which project onto \Sym(D)$. The question arises in model theory as a problem about finite covers, but here we formulate and solve it in algebraic terms.

math.GR↗
arXiv2011

The geometry of Hrushovski constructions, I. The uncollapsed case

An intermediate stage in Hrushovski's construction of flat strongly minimal structures in a relational language L produces omega-stable structures of rank omega. We analyze the pregeometries given by forking on the regular type of rank omega 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.

math.LO↗