Partial associativity and rough approximate groups

Suppose that a binary operation $$\circ $$ ∘ on a finite set X is injective in each variable separately and also associative. It is easy to prove that $$(X,\circ )$$ ( X , ∘ ) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples $$(x,y,z)\in X^3$$ ( x , y , z ) ∈ X 3 satisfy the equation $$x\circ (y\circ z)=(x\circ y)\circ z$$ x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z . Other results in additive combinatorics would lead one to expect that there must be an underlying ‘group-like’ structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.

COSET PRESSURE WITH SUB-ADDITIVE POTENTIALS

The goal of this paper is to define the coset topological pressure for sub-additive potentials via separated sets on a compact metric group. Analogues of basic properties for topological pressure hold. This study also reveals a variational principle for the coset topological pressure. The process of the proof is quite similar to that of Cao, Feng and Huang's approximations, but the analysis needs more techniques of ergodic theory and topological dynamics.

Measures with unique tangent measures in metric groups

We show that a Radon measure on a locally compact metric group with natural dilations has almost everywhere a unique tangent measure if and only if it has almost everywhere a Haar measure of a closed dilation invariant subgroup as its unique tangent measure.

The Livšic periodic point theorem for non-abelian cocycles

For hyperbolic systems and for Hölder cocycles with values in a compact metric group, we extend Livšic's periodic point characterisation of coboundaries. Here we show that two such cocycles are cohomologous when their respective ‘weights’ (of closed orbits) coincide. When it is only assumed that they are conjugate, one of the cocycles must (in general) be modified by an isomorphism (which stabilises conjugacy classes) to obtain cohomology. When the group is Lie and when a transitivity condition is satisfied, conjugacy of weights ensures that the cocycles are cohomologous with respect to a finitely extended group.