Expanders, rank and graphs of groups

By Marc Lackenby. 2004

84 citations recorded by the source corpus.

LetG be a finitely presented group, and let {Gi} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1.Gi is an amalgamated free product or HNN extension, for infinitely manyi; 2. the Cayley graphs ofG/Gi (with respect to a fixed finite set of generators forG) form an expanding family; 3. infi(d(Gi)−1)/[G:Gi]=0, whered(Gi) is the rank ofGi. The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/10840cd1cc9f24638f07bfceade31bdc9e188e42

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Some topics in the dynamics of group actions on rooted trees (2011, methodology): “Another circle of questions related to the study of actions on rooted trees is the study of infinite decreasing chains of finite-index subgroups in residually finite groups, in particular, the study of the rank gradient of these subgroups [121, 5].”
The rank gradient from a combinatorial viewpoint (2007, methodology): “The chain is normal if all Γn are normal in Γ.”
Rank gradient, cost of groups and the rank versus Heegaard genus problem (2007, methodology): “This notion has been introduced by Lackenby [17].”
Nilprogressions and groups with moderate growth (2015, background): “nown whether or not the converse holds in the category of ﬁnitely presented groups. In connection with Thurston’s virtual ﬁrst Betti number conjecture regarding the topology of 3- manifolds, Lackenby [34, 35, 36] studied various extra conditions of an algebraic or topological nature on a ﬁnitely pre”
Expander graphs in pure and applied mathematics (2011, background): “uitable nite cover Y of a 2-dimensional complex Xwith ˇ 1 (X) = can be decomposed as Y = Y 1 [Y 2 in a non-trivial way that will enable to apply van Kampen Theorem to deduce that ˇ 1 (Y) splits - see [La1] for details. As we will see in the next chapter, it is of great importance in the theory of 3-”