Intersection-saturated groups without free subgroups

A group $G$ is said to be intersection-saturated if for every strictly positive integer $n$ and every map $c\colon \mathcal{P}(\{1,\dots, n\})\setminus \emptyset \rightarrow \{0,1\}$, one can find subgroups $H_1,\dots, H_n\leq G$ such that for every non-empty subset $I\subseteq \{1,\dots, n\}$, the intersection $\bigcap_{i\in I}H_i$ is finitely generated if and only if $c(I)=0$. We obtain a new criterion for a group to be intersection-saturated based on the existence of arbitrarily high direct powers of a subgroup admitting an automorphism with a non-finitely generated set of fixed points. We use this criterion to find new examples of intersection-saturated groups, including Thompson's groups and the Grigorchuk group. In particular, this proves the existence of finitely presented intersection-saturated groups without non-abelian free subgroups, thus answering a question of Delgado, Roy and Ventura.Comment: 7 page

All p-local finite groups of rank two for odd prime p

In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analisis of the possible saturated fusion systems in terms of the outer automorphism group ant the proper F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3

Groups in NTP2

We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory

Discrete subgroups of locally definable groups

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group G in this category is a cover of a definable group. We prove that this is the case under a natural convexity assumption inspired by the same authors, which in fact gives a necessary and sufficient condition. The proof is based on the study of the zero-dimensional compatible subgroups of G. Given a locally definable connected group G (not necessarily definably generated), we prove that the n-torsion subgroup of G is finite and that every zero-dimensional compatible subgroup of G has finite rank. Under a convexity hypothesis we show that every zero-dimensional compatible subgroup of G is finitely generated.Comment: Final version. 17 pages. To appear in Selecta Mathematic