A quasi-isometric embedding theorem for groups

We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, F{\o}lner functions, and elementary classes of amenable groups

Extension of information geometry for modelling non-statistical systems

In this dissertation, an abstract formalism extending information geometry is introduced. This framework encompasses a broad range of modelling problems, including possible applications in machine learning and in the information theoretical foundations of quantum theory. Its purely geometrical foundations make no use of probability theory and very little assumptions about the data or the models are made. Starting only from a divergence function, a Riemannian geometrical structure consisting of a metric tensor and an affine connection is constructed and its properties are investigated. Also the relation to information geometry and in particular the geometry of exponential families of probability distributions is elucidated. It turns out this geometrical framework offers a straightforward way to determine whether or not a parametrised family of distributions can be written in exponential form. Apart from the main theoretical chapter, the dissertation also contains a chapter of examples illustrating the application of the formalism and its geometric properties, a brief introduction to differential geometry and a historical overview of the development of information geometry.Comment: PhD thesis, University of Antwerp, Advisors: Prof. dr. Jan Naudts and Prof. dr. Jacques Tempere, December 2014, 108 page

Covering spaces of 3-orbifolds

Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime p. This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of O_i grows linearly in the covering degree. The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about p-adic analytic pro-p groups. This has several consequences. Firstly, the fundamental group of O has at least exponential subgroup growth. Secondly, the covers {O_i} have positive Heegaard gradient. Thirdly, we use it to show that a group-theoretic conjecture of Lubotzky and Zelmanov would imply that O has large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3-orbifold with non-empty singular locus has large fundamental group. Many of the above results apply also to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with non-empty singular locus. This includes all closed orientable hyperbolic 3-manifolds with rank two fundamental group, and all arithmetic 3-manifolds.Comment: 26 pages. Version 3 has only minor changes from versions 1 and 2. To appear in Duke Mathematical Journa