: We consider diagram groups as defined by Guba and Sapir [Mem. Amer. Math. Soc. 130 (1997)]. A diagram group G acts on the associated cube complex K by isometries. It is known that if a cube complex L is of a finite dimension, then every isometry g of L is semi-simple: is attained. It was conjectured by Farley that in the case of a diagram group G the action of G on the associated cube complex K is by semisimple isometries also when K has an infinite dimension. In this paper we give a counter-example to Farley's conjecture by showing that R. Thompson's group F, considered as a diagram group, has some elements which act as parabolic (not semi-simple) isometries on the associated cube complex.

We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.

Locally strictly convex surfaces in four-dimensional affine space are studied from a perspective of the affine structure invented by Nuño-Ballesteros and Sánchez, which is especially suitable in convex geometry. The surfaces that are embedded in locally strictly convex hyperquadrics are classified under assumptions that the second fundamental form is parallel with respect to the induced connection and the normal connection is compatible with a metric on the transversal bundle. Both connections are induced by a canonical transversal plane bundle, which is defined by certain symmetry conditions. The obtained surfaces are always products of an ellipse and a conical planar curve.

A discrete group G is amenable if there exists a finitely additive probability measure on G which is invariant under left translations and is defined on all subsets of G. It is proved that if the group is generated by two elements and is amenable then there are words being relators whose most of the consecutive pairs of the letters belong to a certain four-element set of pairs. This fact is applied to reproving non-amenability of a braid group. The same group provides an example showing that such type of condition is not sufficient for amenabilty.

We give new properties of the minimal growth function for uniformly amenable abelian groups in which orders of elements are not uniformly bounded.

For real finite-dimensional vector spaces V, W we call a bilinear symmetric mapping h : V × V → W non-degenerate if the components of h with respect to a certain basis are linearly independent and non-degenerate. We say that a symmetric trilinear mapping C : V × V × V → W is divisible by h if there exists a linear form α such that C(v, v, v) = α(v)h(v, v) for every v ∈ V. In affine differential geometry of affine immersions h is the second fundamental form and C – the cubic form of the immersion. The immersion has pointwise planar normal sections if h(v, v) ∧ C(v, v, v) = 0 for every tangent vector v. We prove that it implies that C is divisible by h if h is non-degenerate and the codimension is greater than two. For immersions with Wiehe's or Sasaki's choice of transversal bundles divisibility of C by h implies vanishing of C.