Tobias Hartnick

Biharmonic functions on groups and limit theorems for quasimorphisms along random walks

MICHAEL BJORKLUND AND TOBIAS HARTNICK¨Abstract. We show for very general classes of measures on locally compact secondcountable groups that every Borel measurable quasimorphism is at bounded distancefrom a quasi-biharmonic one. This allows us to deduce non-degenerate central limit the-orems and laws of the iterated logarithm for such quasimorphisms along regular randomwalks on topological groups using classical martingale limit theorems of Billingsley andStout. For quasi-biharmonic quasimorphism on countable groups we also obtain integralrepresentations using martingale convergence.

Reconstructing quasimorphisms from associated partial orders and a question of Polterovich

We show that every continuous homogeneous quasimorphism on a finite-dimensional 1-connected simple Lie group arises as the relative growth of any continuous bi-invariant partial order on that group. More generally we show, that an arbitrary homogeneous quasimorphism can be reconstructed as the relative growth of a partial order subject to a certain sandwich condition. This provides a link between invariant orders and bounded cohomology and allows the concrete computation of relative growth for finite dimensional simple Lie groups as well as certain infinite-dimensional Lie groups arising from symplectic geometry.

BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I

We present a new technique that employs partial dierential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous bounded cohomology of SL(2;R) in degree four, establishing a special case of a conjecture of Monod.

Aperiodic order and spherical diffraction, I: auto‐correlation of regular model sets

We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on

Two-spherical topological Kac–Moody groups are Kazhdan

\mathbb R^n

On order-preserving representations

. This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hofs classical definition beyond the class of amenable groups, and prov ide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.

Perturbations of the Spence–Abel equation and deformations of the dilogarithm function

Abstract In this note, we prove that a two-spherical Kac–Moody group over a local field endowed with the Kac–Peterson topology enjoys Kazhdans property (T).

Final group topologies, Kac-Moody groups and Pontryagin duality

In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermitian type are faithful with discrete image and that the set of order preserving representations is closed in the representation variety. For Lie groups of Hermitian type whose associated symmetric space is of tube type we give a geometric characterization of these representations in terms of the causal structure on the Shilov boundary.

Final group topologies, Phan systems and Pontryagin duality

We analyze existence, uniqueness and regularity of solutions for perturbations of the Spence–Abel equation for the Rogers’ dilogarithm. As an application we deduce a version of Hyers–Ulam stability for the Spence–Abel equation. Our analysis makes use of a well-known cohomological interpretation of the Spence–Abel equation and is based on our recent results on continuous bounded cohomology of