Abstract. We study biinvariant word metrics on groups. We provide an efficient algorithm for computing the biinvariant word norm on a finitely generated free group and we construct an isometric embedding of a locally compact tree into the biinvariant Cayley graph of a nonabelian free group. We investigate the geometry of cyclic subgroups. We observe that in many classes of groups cyclic subgroups are either bounded or detected by homogeneous quasimorphisms. We call this property the bqdichotomy and we prove it for many classes of groups of geometric origin.

Let S be a compact oriented surface. We construct homogeneous quasimorphisms on Diff(S, area), on Diff0(S, area) and on Ham(S) generalizing the constructions of Gambaudo-Ghys and Polterovich.We prove that there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S, area), on Diff0(S, area) and on Ham(S) whose absolute values bound from below the topological entropy. In case when S has a positive genus, the quasimorphisms we construct on Ham(S) are C 0 -continuous.We define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric on Ham(S) is unbounded.

This paper presents a rare case of facial soft tissue infection caused by the bacterial strain of Rothia mucilaginosa. Odontogenic background of infection and initial clinical presentation suggested the presence of typical bacterial flora and uncomplicated course of treatment. However, despite surgical intervention and broad-spectrum antibiotic therapy, the expected improvement of a clinical status was not achieved. Only detailed bacteriological examination allowed to establish a bacterial pathogen and start a targeted antibiotic therapy. The unusual clinical course was monitored by imaging CT examination and further surgical interventions. A significant improvement was obtained in the third week of hospitalization and further antibiotic therapy was continued by means of outpatient treatment. Rothia mucilaginosa infection together with dental intervention is a rare case, since most of the reports in the literature concern the patients with decreased immunity. In such patients, the most common areas of infection were: the peritoneum, lung tissue and meningeal spaces of the brain and the presence of a foreign body.

Let Fn be the free group on n generators and Γg the surface group of genus g. We consider two particular generating sets: the set of all primitive elements in Fn and the set of all simple loops in Γg. We give a complete characterization of distorted and undistorted elements in the corresponding Aut-invariant word metrics. In particular, we reprove Stallings theorem and answer a question of Danny Calegari about the growth of simple loops. In addition, we construct infinitely many quasimorphisms on F2 that are AutpF2q-invariant. This answers an open problem posed by Miklós Abért.2010 Mathematics Subject Classification. 57.

ABSTRACT. We give a homological construction of aperiodic tiles for certain open Riemannian surfaces admitting actions of Grigorchuk groups of intermediate growth.Let X be a noncompact Riemannian manifold. A set of tiles for X is a triple {T , W, o}, where T is a finite collection of compact polygons with boundary (tiles), each with distinguished connected faces, W is a collection of all faces of T and o : W → W is an opposition (matching) function. A tiling of X is a cover X = ∪ α X α , where each X α is isometric to a tile in W, every non-empty intersection of two distinct pieces is identified with faces w α and w β of the corresponding tiles and satisfies o(w α ) = w β . A tiling is (weakly) aperiodic if no group acting on X cocompactly by isometries preserves the tiling. An aperiodic set of tiles of X is a set of tiles, which admits only aperiodic tilings. Classical examples include aperiodic tiles of the Euclidean spaces, such as Penrose tiles of the plane.Let M → M be an infinite covering of a closed manifold. In [BW] Block and Weinberger constructed aperiodic tiles for M when the covering group is nonamenable. Their construction relies on the fact that, for such a group, its uniformly finite homology with coefficients in Z is trivial in degree 0. Unfortunately, this homology group is highly nontrivial for amenable groups. Apart from the Euclidean spaces, there are virtually no constructions of aperiodic tiles for amenable manifolds. Recently the second author with S. Weinberger constructed aperiodic tiles for the real Heisenberg group by a different method (unpublished).In this note we return to the original method of [BW] and the vanishing of uniformly finite homology in degree 0. We construct, using homology with torsion coefficients, aperiodic tiles for manifolds equipped with proper cocompact actions of the Grigorchuk group or certain other groups of intermediate growth It is well known that such a manifolds are amenable (i.e., regularly exhaustible).