| EPSRC Reference: |
EP/H045112/1 |
| Title: |
Rank gradient of groups |
| Principal Investigator: |
Nikolov, Dr N |
| Other Investigators: |
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| Researcher Co-Investigators: |
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| Project Partners: |
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| Department: |
Mathematics |
| Organisation: |
Imperial College London |
| Scheme: |
Standard Research |
| Starts: |
01 April 2011 |
Ends: |
01 September 2012 |
Value (£): |
336,171
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| EPSRC Research Topic Classifications: |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Related Grants: |
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| Panel History: |
| Panel Date | Panel Name | Outcome |
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04 Mar 2010
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Mathematics Prioritisation Panel
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Announced
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Summary on Grant Application Form |
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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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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.imperial.ac.uk |