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

  title={Biharmonic functions on groups and limit theorems for quasimorphisms along random walks},
  author={Michael Bjorklund and Tobias Hartnick},
  journal={Geometry \& Topology},
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… 

Quasimorphisms, random walks, and transient subsets in countable groups

We study the interrelations between the theory of quasimorphisms and the theory of random walks on groups, and establish the following transience criterion for subsets of groups: if a subset of a

Central limit theorems for counting measures in coarse negative curvature

We establish central limit theorems for an action of a group $G$ on a hyperbolic space $X$ with respect to the counting measure on a Cayley graph of $G$. Our techniques allow us to remove the usual

Deviation inequalities for random walks

We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of

Simultaneous construction of hyperbolic isometries

Given isometric actions by a group G on finitely many \delta-hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in G that is hyperbolic for

Statistics and compression of scl

Abstract We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting non-degenerately on

Banach limits: extreme properties, invariance and the Fubini theorem

A Banach limit on the space of all bounded real sequences is a positive normalized linear functional that is invariant with respect to the shift. The paper studies such properties of Banach limits as

Random rigidity in the free group

We prove a rigidity theorem for the geometry of the unit ball in random subspaces of the scl norm in B_1^H of a free group. In a free group F of rank k, a random word w of length n (conditioned to

Higher Teichmüller Spaces: from SL(2,R) to other Lie groups

The first part of this paper surveys several characterizations of Teichm\"uller space as a subset of the space of representation of the fundamental group of a surface into PSL(2,R). Special emphasis

Extremality of the rotation quasimorphism on the modular group

For any element A of the modular group PSL(2,Z), it follows from work of Bavard that scl(A) is greater than or equal to rot(A)/2, where scl denotes stable commutator length and rot denotes the

A central limit theorem for the degree of a random product of rational surface maps

We prove a central limit theorem for the algebraic and dynamical degrees of a random composition of dominant rational maps of P^2.



Double ergodicity of the Poisson boundary and applications to bounded cohomology

AbstractWe prove that the Poisson boundary of any spread out non-degenerate symmetric randomwalk on an arbitrary locally compact second countable group G is doubly $\mathcal{M}$sep-ergodic with

Lengths, quasi-morphisms and statistics for free groups

There is considerable interest in spaces of length functions defined on a free group F on k � 2 generators (or on its set of conjugacy classes). For example, the Culler-Vogtmann Outer space [13] has

Combable functions, quasimorphisms, and the central limit theorem

Abstract A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable

Bounded cohomology of lattices in higher rank Lie groups

We prove that the natural map Hb2(Γ)?H2(Γ) from bounded to usual cohomology is injective if Γ is an irreducible cocompact lattice in a higher rank Lie group. This result holds also for nontrivial

The filter Dichotomy and Medial Limits

It is shown that the Filter Dichotomy implies that there are no medial limits, and Christensen and Mokobodzki independently showed that the Continuum Hypothesis implies the existence of medial limits.

The Lindeberg-Lévy theorem for martingales

The central limit theorem of Lindeberg [7] and Levy [3] states that if {mi, m2, ■ ■ • } is an independent, identically distributed sequence of random variables with finite second moments, then the

Noncommuting random products

Introduction. Let Xy,X2, ■■-,X„,--be a sequence of independent real valued random variables with a common distribution function F(x), and consider the sums Xy + X2 + ■■• + X„. A fundamental theorem

Bi-harmonic functions on groups

Nous montron l'equivalence essentielle de deux definitions naturelles des fonctions biharmoniques sur les groupes denombrables et etablissons des criteres d'absence de fonctions biharmoniques

Doctoral thesis

In this thesis, the existence and uniqueness of gradient trajectories near an A2singularity are analysed and it is proved that the two Lagrangian vanishing cycles associated to these critical points intersect transversally in exactly one point in all regular fibres along a straight line.