23,308 research outputs found
One brick at a time: a survey of inductive constructions in rigidity theory
We present a survey of results concerning the use of inductive constructions
to study the rigidity of frameworks. By inductive constructions we mean simple
graph moves which can be shown to preserve the rigidity of the corresponding
framework. We describe a number of cases in which characterisations of rigidity
were proved by inductive constructions. That is, by identifying recursive
operations that preserved rigidity and proving that these operations were
sufficient to generate all such frameworks. We also outline the use of
inductive constructions in some recent areas of particularly active interest,
namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar
frameworks. We summarize the key outstanding open problems related to
inductions.Comment: 24 pages, 12 figures, final versio
Amenability and geometry of semigroups
We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Følner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Følner condition in terms of the existence of weak Følner sets satisfying a local injectivity condition on the relevant translation action of the semigroup
Automorphisms of graph products of groups from a geometric perspective
This article studies automorphism groups of graph products of arbitrary
groups. We completely characterise automorphisms that preserve the set of
conjugacy classes of vertex groups as those automorphisms that can be
decomposed as a product of certain elementary automorphisms (inner
automorphisms, partial conjugations, automorphisms associated to symmetries of
the underlying graph). This allows us to completely compute the automorphism
group of certain graph products, for instance in the case where the underlying
graph is finite, connected, leafless and of girth at least . If in addition
the underlying graph does not contain separating stars, we can understand the
geometry of the automorphism groups of such graph products of groups further:
we show that such automorphism groups do not satisfy Kazhdan's property (T) and
are acylindrically hyperbolic. Applications to automorphism groups of graph
products of finite groups are also included. The approach in this article is
geometric and relies on the action of graph products of groups on certain
complexes with a particularly rich combinatorial geometry. The first such
complex is a particular Cayley graph of the graph product that has a
quasi-median geometry, a combinatorial geometry reminiscent of (but more
general than) CAT(0) cube complexes. The second (strongly related) complex used
is the Davis complex of the graph product, a CAT(0) cube complex that also has
a structure of right-angled building.Comment: 36 pages. The article subsumes and vastly generalises our preprint
arXiv:1803.07536. To appear in Proc. Lond. Math. So
Recognising Multidimensional Euclidean Preferences
Euclidean preferences are a widely studied preference model, in which
decision makers and alternatives are embedded in d-dimensional Euclidean space.
Decision makers prefer those alternatives closer to them. This model, also
known as multidimensional unfolding, has applications in economics,
psychometrics, marketing, and many other fields. We study the problem of
deciding whether a given preference profile is d-Euclidean. For the
one-dimensional case, polynomial-time algorithms are known. We show that, in
contrast, for every other fixed dimension d > 1, the recognition problem is
equivalent to the existential theory of the reals (ETR), and so in particular
NP-hard. We further show that some Euclidean preference profiles require
exponentially many bits in order to specify any Euclidean embedding, and prove
that the domain of d-Euclidean preferences does not admit a finite forbidden
minor characterisation for any d > 1. We also study dichotomous preferencesand
the behaviour of other metrics, and survey a variety of related work.Comment: 17 page
Geometric Property (T)
This paper discusses `geometric property (T)'. This is a property of metric
spaces introduced in earlier work of the authors for its applications to
K-theory. Geometric property (T) is a strong form of `expansion property': in
particular for a sequence of finite graphs , it is strictly stronger
than being an expander in the sense that the Cheeger constants
are bounded below.
We show here that geometric property (T) is a coarse invariant, i.e. depends
only on the large-scale geometry of a metric space . We also discuss the
relationships between geometric property (T) and amenability, property (T), and
various coarse geometric notions of a-T-menability. In particular, we show that
property (T) for a residually finite group is characterised by geometric
property (T) for its finite quotients.Comment: Version two corrects some typos and a mistake in the proof of Lemma
8.
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