A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable

This is an old paper put here for archeological purposes. It is proved that a finite-dimensional Lie algebra over a field of characteristic p>5, that can be written as a vector space (not necessarily direct) sum of two nilpotent subalgebras, is solvable. The same result (but covering also the cases of low characteristics) was established independently by V. Panyukov (Russ. Math. Surv. 45 (1990), N4, 181-182), and the homological methods utilized in the proof were developed later in arXiv:math/0204004. Many inaccuracies in the English translation are corrected, otherwise the text is identical to the published version.Comment: v2: minor change

Presentations: from Kac-Moody groups to profinite and back

We go back and forth between, on the one hand, presentations of arithmetic and Kac-Moody groups and, on the other hand, presentations of profinite groups, deducing along the way new results on both

A Principled Approach to Analyze Expressiveness and Accuracy of Graph Neural Networks

Graph neural networks (GNNs) have known an increasing success recently, with many GNN variants achieving state-of-the-art results on node and graph classification tasks. The proposed GNNs, however, often implement complex node and graph embedding schemes, which makes challenging to explain their performance. In this paper, we investigate the link between a GNN's expressiveness, that is, its ability to map different graphs to different representations, and its generalization performance in a graph classification setting. In particular , we propose a principled experimental procedure where we (i) define a practical measure for expressiveness, (ii) introduce an expressiveness-based loss function that we use to train a simple yet practical GNN that is permutation-invariant, (iii) illustrate our procedure on benchmark graph classification problems and on an original real-world application. Our results reveal that expressiveness alone does not guarantee a better performance, and that a powerful GNN should be able to produce graph representations that are well separated with respect to the class of the corresponding graphs

Finite covers of random 3-manifolds

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) -> Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman's theorem that the action of the mapping class group on the SU(2) character variety is ergodic.Comment: 60 pages; v2: minor changes. v3: minor changes; final versio