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Abstract: Given a subroup G of a stable (in the model-theoric sense) group Γ, in particular when Γ is a group of finite Morley rank, the traces on G of the definable subsets of Γ have a remarkable property: if the definable closure of G is connected, they are either supergeneric, or supergenerically complemented, in the sense of the definition given at the very beginning of this paper. An example of this situation is provided by the linear groups: for some n, G is a subgroup of , where K is a field that we may take algebraically closed; the definable sets in the sense of are its constructible subsets, i.e. the boolean combinations of a finite number of its Zariski closed subsets. For any group G, the supergeneric subsets of G form a filter of large sets, which, to my best knowledge, is defined here for the first time. This paper undertakes the study of supergenericity in a general context, with no hypotheses of a model-theoric nature, but with a special attention given to the very specific properties of genericity possessed by the definable subsets of a stable group. It can be read without any knowledge of Logic, provided that one is ready to skip the proofs of the theorems showing precisely that these definable sets have these properties. [Copyright &y& Elsevier]

Let N be the set of positive integers and A a subset of N. For n∈N, let p(A,n) denote the number of partitions of n with parts in A. In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any N∈N and B⊂{1,2,…,N}, there is a unique set A=A0(B,N), such that p(A,n) is even for n>N. Soon after, Ben Saı¨d and Nicolas (Acta Arith. 106 (2003) 183) considered σ(A,n)=∑d∣n,d∈Ad, and proved that for all k≥0, the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. In this paper, we generalise the above works for any formal power series f in F2[z] with f(0)=1, by constructing a set A such that the generating function fA of A is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in F2[z] with P(0)=Q(0)=1, then for all k≥0 the sequence (σ(A,2kn) mod 2k+1)n≥1 is periodic on n. [Copyright &y& Elsevier]

Abstract: In this paper, we first characterize the subsets of the Bruhat–Tits tree of , K a complete valued field, that are the sets of fixed points of a subgroup G of . When G is irreducible, the are the “strips” in the tree. We then evaluate the form of the strip , in particular in terms of algebraic invariants of the group G. We give two applications to representation theory, one about a new generalization of Ribet's lemma on extensions, the other about the trace-convergence of a sequence of representations. [Copyright &y& Elsevier]

This paper presents an axiomatic description in a base free way of infinite root systems as those found in the study of Kac–Moody–Borcherds' algebras. In 1979, R. Moody [Adv. Math. 33 (1979) 144] posed the problem of finding such a description. In [Comm. Algebra 23 (1995) 4791], J.G. Bliss has already presented two answers to this question with his notions of “geometric root systems” and “rational root systems.” This new answer to Moody's question fits into the framework of the earlier axiomatic theory of the « syste`mes ge´ne´rateurs de racines » developed in [N. Bardy, Me´m. Soc. Math. Fr. (N.S.) 65 (1996)]. [Copyright &y& Elsevier]