EPSRC Reference: 
EP/H045112/1 
Title: 
Rank gradient of groups 
Principal Investigator: 
Nikolov, Dr N 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Imperial College London 
Scheme: 
Standard Research 
Starts: 
01 April 2011 
Ends: 
01 September 2012 
Value (£): 
336,171

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 
Panel Date  Panel Name  Outcome 
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 3manifolds, 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 3manifolds: 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 3dimensional Topology, Group theory, Ergodic actions and Arithmetic Groups.

Key Findings 
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Potential use in nonacademic contexts 
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Impacts 
Description 
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Summary 

Date Materialised 


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Project URL: 

Further Information: 

Organisation Website: 
http://www.imperial.ac.uk 