相关论文: Rank gradient, cost of groups and the rank versus …

We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and…

Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.

In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating…

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove…

We analyze the rank gradient of finitely generated groups with respect to sequences of subgroups of finite index that do not necessarily form a chain, by connecting it to the cost of p.m.p. actions. We generalize several results that were…

The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that…

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of…

We construct first examples of infinite finitely generated residually finite torsion groups with positive rank gradient. In particular, these groups are non-amenable. Some applications to problems about cost and $L^2$-Betti numbers are…

We prove a finiteness theorem for subgroups of bounded rank in hyperbolic $3$-manifold groups. As a consequence, we show that every bounded rank covering tower of closed hyperbolic $3$-manifolds is a tower of finite covers associated to a…

Residual finiteness growth gives an invariant that indicates how well-approximated a finitely generated group is by its finite quotients. We briefly survey the state of the subject. We then improve on the best known upper and lower bounds…

We prove a conjecture of Helfgott on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of $\mathrm{GL}_n(\mathbf{F})$ for any field $\mathbf{F}$ such that $|A^3| \leq…

This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble…

We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower…

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the…

A new bound for the rank of the intersection of finitely generated subgroups of a free group is given, formulated in topological terms, and very much in the spirit of Stallings. The bound is a contribution to (although unfortunately not a…

The residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. In this paper, we construct groups with arbitrarily large residual finiteness growth. We also demonstrate a new relationship…

We study the growth of product sets in some finite three-dimensional matrix groups. In particular, we prove two results about the group of $2\times 2$ upper triangular matrices over arbitrary finite fields: a product set estimate using…

We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced…

It is proven that an infinite finitely generated group cannot be elementarily equivalent to an ultraproduct of finite groups of a given Pr\"ufer rank. Furthermore, it is shown that an infinite finitely generated group of finite Pr\"ufer…

In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1981), no. 1-2, pages 25-45; MR 617288) A. Lubotzky and L. van den Dries raise the question whether a finitely generated, residually finite group is…