Conservative median algebras and semilattices

We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median algebras and certain topological structures, we obtain descriptions of the median-preserving mappings between products of finitely many chains

Transit functions on graphs (and posets)

The notion of transit function is introduced to present a unifying approachfor results and ideas on intervals, convexities and betweenness in graphs andposets. Prime examples of such transit functions are the interval function I andthe induced path function J of a connected graph. Another transit function isthe all-paths function. New transit functions are introduced, such as the cutvertextransit function and the longest path function. The main idea of transitfunctions is that of ‘transferring’ problems and ideas of one transit functionto the other. For instance, a result on the interval function I might suggestsimilar problems for the induced path function J. Examples are given of howfruitful this transfer can be. A list of Prototype Problems and Questions forthis transferring process is given, which suggests many new questions and openproblems.graph theory;betweenness;block graph;convexity;distance in graphs;interval function;path function;induced path;paths and cycles;transit function;types of graphs

Kazhdan and Haagerup properties from the median viewpoint

We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on $L^p$-spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson-Steger.Comment: final versio

Betweenness algebras

We introduce and study a class of betweenness algebras-Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators of possibility and of suffciency.Comment: 26 pages, 2 figure

Steps in the Representation of Concept Lattices and Median Graphs

International audienceMedian semilattices have been shown to be useful for dealing with phylogenetic classication problems since they subsume median graphs, distributive lattices as well as other tree based classica-tion structures. Median semilattices can be thought of as distributive ∨-semilattices that satisfy the following property (TRI): for every triple x, y, z, if x ∧ y, y ∧ z and x ∧ z exist, then x ∧ y ∧ z also exists. In previous work we provided an algorithm to embed a concept lattice L into a dis-tributive ∨-semilattice, regardless of (TRI). In this paper, we take (TRI) into account and we show that it is an invariant of our algorithmic approach. This leads to an extension of the original algorithm that runs in polynomial time while ensuring that the output is a median semilattice

Large-scale rigidity properties of the mapping class groups

We study the coarse geometry of the mapping class group of a compact orientable surface. We show that, apart from a few low-complexity cases, any quasi-isometric embedding of a mapping class group into itself agrees up to bounded distance with a left multiplication. In particular, such a map is a quasi-isometry. This is a strengthening of the result of Hamenst¨adt and of Behrstock, Kleiner, Minsky and Mosher that the mapping class groups are quasi-isometrically rigid. In the course of proving this, we also develop the general theory of coarse median spaces and median metric spaces with a view to applications to Teichm¨uller space, and related spaces