Geometric Group Theory
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
Just as numbers are the mathematical objects that measure size, groups are the mathematical objects that one needs to describe the symmetries of any mathematical object: no matter what category of objects one is working with, the symmetries (automorphisms) of each object form a group.Having abstracted the notion of a group, it makes sense to study groups as important objects in their own right; thus one has group theory. But in order to elucidate the true nature of an abstract group, one often wants to realise it as the group of automorphisms of specific objects and to use the structure of those objects in order to elucidate the structure of the group. This idea lies at the heart of geometric group theory: typically, given a group, one seeks objects rich in geometric (or other) structure and studies the realisation(action) of the group on this object, using the pwoerful tools of geometry and topology.Conversely, one tries to solve problems in geometry and topology by analysing the groups associated to the spaces at hand, or by encoding well-understood phenomena from group theory into geometric objects.The research to be undertaken here involves both of these approaches:an understanding of the universe of finitely presented groups is sought, largely through an understanding of the geometry and complexity of groups and their actions; on the other hand, attacks on important geometric and topological problems are mounted via group theory.
Organisations
People |
ORCID iD |
Martin Robert Bridson (Principal Investigator) |
Publications
Aliverdiev A
(2008)
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Anderson J
(2016)
Relative shapes of thick subsets of moduli space
in American Journal of Mathematics
Anderson J
(2013)
Relative shapes of thick subsets of moduli space
Anderson James W.
(2016)
RELATIVE SHAPES OF THICK SUBSETS OF MODULI SPACE
in AMERICAN JOURNAL OF MATHEMATICS
Baring L
(2011)
Local and global superconductivity in bismuth
in Low Temperature Physics
Bridson M
(2012)
On the dimension of CAT(0) spaces where mapping class groups act
in Journal für die reine und angewandte Mathematik (Crelles Journal)
Bridson M
(2013)
Geometry of Riemann Surfaces
Bridson M
(2013)
Constructing presentations of subgroups of right-angled Artin groups
in Geometriae Dedicata
Bridson M
(2009)
Extrinsic versus intrinsic diameter for Riemannian filling-discs and van Kampen diagrams
in Journal of Differential Geometry
Bridson M
(2013)
Actions of arithmetic groups on homology spheres and acyclic homology manifolds
in Mathematische Zeitschrift
Description | Diverse and significant advances were made in the understanding of geometric aspects of group theory and associated areas of mathematics. Highlights include: (1) a definitive structure theory for finitely presented residually free groups; (2) an associated general theory of subdirect products and their finiteness properties; (3) many advances in the understanding of automorphism groups of free groups, mapping class groups and their actions, including rigidity theorems for maps between these types of groups, the theorem that Out(F_n) cannot act non-trivially on a contractible manifold of dimension less than n, and constraints on the dimension of CAT(0) spaces on which these groups can act, as well as the determination of Dehn functions and commensurability results for subgroups; (4) a resolution of the conjecture that higher rank lattices can only act on free groups via finite quotients; (5) major breakthroughs in the understanding of filling invariants for finitely presented groups and manifolds (in particular the nature of word problems and isoperimetric problems); the denseness of isoperimetric spectra in all dimensions was established; (6) proving that every free-by-cyclic group satisfies a quadratic isoperimetric inequality; (7) comprehensive fixed point theorems; (8) the initiation of the study of decision problems in the context of finitely presented profinite groups, advances in the understanding of (non)invariants of (weak and strong) profinite genus, and laying the foundations for the study of Grothendieck rigidity. |
Exploitation Route | They are widely disseminated in my various writings. Advances in the mainstream of fundamental mathematics have, throughout history and most particularly the last century, advanced the social and economic development of society in unexpected and often spectacular ways, but the timescales and origin of the application of the deepest ideas are impossibly difficult to predict. I am unaware of any non-academic applications of my work for the moment, but hope very much that these will emerge in my lifetime. All of the results obtained in the course of this fellowship were published in mathematical journals of international standing |
Sectors | Digital/Communication/Information Technologies (including Software),Education,Financial Services, and Management Consultancy,Security and Diplomacy,Other |
Description | To further research in fundamental mathematics, across a broad part of the subject -- they are widely cited. There is also the expectation that (on unknown timescales), this will feed into mathematical advances that have a transformative impact through scientific advances that determine the future of mankind. |
First Year Of Impact | 2007 |
Sector | Digital/Communication/Information Technologies (including Software),Education,Security and Diplomacy,Other |
Impact Types | Cultural |
Description | Royal Society Wolfson Research Merit Award |
Amount | £125,000 (GBP) |
Organisation | The Royal Society |
Sector | Charity/Non Profit |
Country | United Kingdom |
Start | 05/2012 |
End | 05/2017 |