The 4-string Braid group B4B_4 has property RD and exponential mesoscopic rank

We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group \Aut(F_2) of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.Comment: reference added, minor correction

Removing chambers in Bruhat-Tits buildings

We introduce and study a family of countable groups constructed from Euclidean buildings by "removing" suitably chosen subsets of chambers

Random groups and nonarchimedean lattices

We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to M. Gromov's well-known constructions and include for example a "density model" for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomization of certain families of lattices in algebraic groups (of rank 2) over local fields

Turbulence, representations, and trace-preserving actions

We establish criteria for turbulence in certain spaces of C*-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,\mu) and on the hyperfinite II_1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,\mu) is generically turbulent.Comment: 27 page

Cracking the Shell of the Vegan Egg Substitute Market.

Veganism has experienced exponential growth in the past decade and is gaining increased consumer attention (Forgrive, 2018). The increased demand for vegan products extends into alternative products, as consumers adapt their lifestyle yet want an easy substitute for their favorite foods and researchers are interested in the economic translation of these consumer choices (Hepting et al, 2014). An exploratory survey was distributed to a southwest Michigan university and through social media which was completed by 302 self-selected American adults. One-way ANOVAs indicated that dietary preference impacted how much people enjoyed vegan substitutes (F(3,298) = 24.64, p \u3c 0.001), dietary preference length did not influence the enjoyment of vegan substitutes (F(3,222) = 2.12, p = 0.099), and dietary preference did not impact issues encountered (taste, consistency, appearance, and cost) with vegan egg substitutes (F(3,298) = 0.096, p = 0.962; F(3,298) = 2.645, p = 0.049; F(3,298) = 0.382, p = 0.766; F(3,298) = 2.765, p = 0.042). The relationship between cooking and vegan egg substitute use is also explored. The colors associated with health, veganism, and eggs are also investigated and, across dietary preferences and consumers, there is a common perception of the colors associated with health, veganism, and eggs