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Details of Grant 

EPSRC Reference: EP/H045112/1
Title: Rank gradient of groups
Principal Investigator: Nikolov, Dr N
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Dept of Mathematics
Organisation: Imperial College London
Scheme: Standard Research
Starts: 01 April 2011 Ends: 01 September 2012 Value (£): 336,171
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
04 Mar 2010 Mathematics Prioritisation Panel Announced
Summary on Grant Application Form
The aim of this project is to investigate a new numerical invariant of infinite groups: rank gradient.Roughly this measures the rate of growth of the generators of normal finite index subgroups in our infinite group. This concept was first introduced by Marc Lackenby and has been very useful in the study of hyperbolic 3-manifolds, in particular the virtually Haken conjecture. On the other hand recently Abert and Nikolov have connected the rank gradient with the notion of cost as used by D. Gaboriau in his study of measurable equivalence of group actions. Results so far already show that the fixed price question from that subject is incompatible with an old question of Waldhousen about the rank and Heegaard genus of hyperbolic 3-manifolds: at least one of these has a negative answer but we don't know which one of the two. More specifically it will be enough to resolve the rank gradient of Kleinian groups. Even achieving this for just one group e.g. PSL(2, Z[i]) will produce spectacular results: it will resolve the above dichotomy and in addition will have implications for the congruence kernel of the Kleinian groups, a subject of interest to Number and Group theorists. To resolve this we need to investigate deeper the connection of the rank gradient of a group with the nature of its action on its completions. This project aims to discover these connections and in the process give answer the above problems spanning 3-dimensional Topology, Group theory, Ergodic actions and Arithmetic Groups.
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Organisation Website: http://www.imperial.ac.uk