On twisted Fourier analysis and convergence of Fourier series on discrete groups

We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.Comment: 35 pages; abridged, revised and update

Gorilla mothers also matter! New insights on social transmission in gorillas (Gorilla gorilla gorilla) in captivity

The present paper describes two distinct behaviors relating to food processing and communication that were observed in a community of five separately housed groups of lowland gorillas (Gorilla gorilla gorilla) in captivity during two study periods one decade apart: (1) a food processing technique to separate wheat from chaff, the so-called puff-blowing technique; and (2) a male display used to attract the attention of visitors, the so-called throw-kiss-display. We investigated (a) whether the behaviors were transmitted within the respective groups; and if yes, (b) their possible mode of transmission. Our results showed that only the food processing technique spread from three to twenty-one individuals during the ten-year period, whereas the communicative display died out completely. The main transmission mode of the puff-blowing technique was the mother-offspring dyad: offspring of puff-blowing mothers showed the behavior, while the offspring of non- puff-blowing mothers did not. These results strongly support the role mothers play in the acquisition of novel skills and vertical social transmission. Furthermore, they suggest that behaviors, which provide a direct benefit to individuals, have a high chance of social transmission while the loss of benefits can result in the extinction of behaviors