Flavonoid Glycosides from the Aerial Part of Lespedeza tomentosa

Chemistry of Natural Compounds - The aerial part of Lespedeza tomentosa (Thunb.) Maxim. growing in southern Primorsky Krai, Russian Federation, yielded 10 identified flavonoid glycosides, eight of...     

A Structural Property Concerning Abstract Commensurability of Subgroups

Further properties of a group introduced by the first authorin 1980 (sometimes called the first Grigorchuk group) are established.These are notably that any two infinite finitely generated subgroupsof have isomorphic subgroups of finite index and that the generalizedword problem for is soluble.     

Weight functions on groups and an amenability criterion for Beurling algebras

The paper is devoted to the study of weights on groups. A connection between weight functions and harmonic functions is established. A relationship between the weight theory on groups with the “Tychonoff property” and the theory of bounded cohomology is presented. It is proved that the Beurling algebraℓ1 (G, ω) constructed for the weightω is amenable if and only if the groupG is amenable and the weightω is equivalent to a multiplicative characterχ:G→ℝ₊. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 448–460, September, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00974 and by the INTAS Foundation under grant No. 94-3420.     

The Uniqueness of the Actions of Certain Branch Groups on Rooted Trees

It is proved that in many cases of interest the actions of groups on rooted trees can be recovered from the structure of the groups. The results apply to most of the groups introduced by the first author and to the Gupta–Sidki groups; they are proved in the wider context of branch groups satisfying two natural conditions.     

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acting on a regular rooted tree acts absolutely non-freely on the boundary of the tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II₁-factor representations, and infinitely many continuous ergodic invariant random subgroups.     

Ergodic decomposition of group actions on rooted trees

We prove a general result about the decomposition into ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. Special attention is paid to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and so-called universal group, are considered in order to demonstrate applications of the theorem.