In this paper, we will consider finitely presented groups that have a finite index subgroup which admits a surjective homomorphism onto a non-abelian free group. Gromov called these groups large [4]. Large groups have particularly nice properties (for example, super-exponential subgroup growth). They also play an important rôle in lowdimensional topology: it is a major conjecture that the funda...

The aim of this lecture is to survey some recent developments in the theory of finite semigroups. More precisely, we shall consider the following problem about pseudovarieties of semigroups: given two pseudovarieties V and W, find a description of their join V ∨W (that is, of the pseudovariety they generate). This question is motivated by the theory of rational languages: it appears in a natura...

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We study the partially-ordered abelian group, F (P; R), generated by a set of generators,P , and inequality relations, R, and its representations in Euclidean space. First F (P; R) is deened by a universal property and then existence is shown proof-theoretically. We then characterize the order structure of F (P; R). The rst main result of the paper utilizes the Marriage Theorem to prove that if...

We prove that if a finitely presented group is one-ended then its asymptotic dimension is bigger than 1. It follows that a finitely presented group of asymptotic dimension 1 is virtually free.

In [3] we investigated finitely presented lattices and the closely related subject of lattices generated by a finite partial lattice. We described a canonical form for the elements of such a lattice and used this to study the covering relation. We showed that there is an effective procedure for finding the covers of any element of a finitely presented lattice. We gave an example of a finitely p...

We construct a finitely presented torsion-free simple group Σ0, acting cocompactly on a product of two regular trees. An infinite family of such groups has been introduced by Burger-Mozes ([2, 4]). We refine their methods and get Σ0 as an index 4 subgroup of a group Σ < Aut(T12)×Aut(T8) presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [4,...

This paper presents a formalization of constructive module theory in the intuitionistic type theory of Coq. We build an abstraction layer on top of matrix encodings, in order to represent finitely presented modules, and obtain clean definitions with short proofs justifying that it forms an abelian category. The goal is to use it as a first step to get certified programs for computing topologica...

We present sufficient conditions for a unary semigroup variety to have no finite basis for its equational theory. In particular, we exhibit a 6-element involutory semigroup which is inherently non-finitely based as a unary semigroup. As applications we get several naturally arising unary semigroups without finite identity bases, for example: the semigroup of all complex 2 × 2-matrices endowed w...

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) ≥ 2 and let Γ be a uniform lattice in G. (a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism. (b) If CH fails, then Γ has 2 ω asymptotic cones up to homeomorphism.