We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given $${n\in N\cup\left\{\omega\right\}}$$ , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.

To interpret interacting particle system style models as social dynamics, suppose each pair {i, j} of individuals in a finite population meet at random times of arbitrary specified rates vij, and update theirstates according to some specified rule. The averaging process has real-valued states and the rule: upon meeting, the values Xi(t-),Xj (t-) arereplaced by 1/2 (Xi(t-) + Xj (t-)), 1/2 (Xi(t-) + Xj (t-)). It is curious this simple process has not been studied very systematically. We provide an expository account of basic facts and open problems.

Given an edge-weighted tree T with n leaves, sample the leaves uniformly at random without replacement and let W_k , 2 ≤ k ≤ n, be the length of the subtree spanned by the first k leaves. We consider the question, "Can T be identified (up to isomorphism) by the joint probability distribution of the random vector (W₂, …, W_n )?" We show that if T is known a priori to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted trees, and certain families with equal weights on all edges such as (k + 1)-valent and rooted k-ary trees for k ≥ 2 and caterpillars.

In the compulsive gambler process there is a finite set of agents who meet pairwise at random times (i and j meet at times of a rate-vij Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other’s money. We introduce this process and describe some of its basic properties. Some properties are rather obvious (martingale structure; comparison with Kingman coalescent) while others are more subtle (an “exchangeable over the money elements" property, and a construction reminiscent of the Donnelly-Kurtz look-down construction). Several directions for possible future research are described. One – where agents meet neighbors in a sparse graph – is studied here, and another – a continuous-space extension called the metric coalescent – is studied in Lanoue (2014).