## GrahamNiblo

Primary position:
Other positions:
Professor of Mathematics

### Background

• MsC Pure Mathematics (Distinction), University of London, 1985
• PhD Pure Mathematics, University of Liverpool, 1988
• Research Assistant, University of Michigan, 1988
• SERC Postdoctoral Fellow, Sussex, 1988-1990
• SERC Postdoctoral Research Assistant, London, 1990-1992
• Lecturer University of Southampton, 1992-2000
• Visiting Professor, Rutgers, 1998
• Senior Lecturer Southampton University, 2000-2005
• Professor of Mathematics, University of Southampton, 2005-
• Head of Pure Mathematics, University of Southampton, 2006-2011
• Leverhulme Research Fellow, 2011-2012
• Visiting Researcher, University of Oxford, 2011
• Head of Mathematics, University of Southampton, 2012-

## Article

Brodzki, Jacek, Niblo, Graham A., Plymen, Roger and Wright, Nick (2014) The local spectrum of the Dirac operator for the universal cover of SL_2(R). Journal of Functional Analysis (Submitted).
Capraro, Valerio, Smyth, Conor, Mylona, Kalliopi and Niblo, Graham A. (2014) Benevolent characteristics promote cooperative behaviour among humans. PLoS ONE, 9, (8)
Sanchez-Garcia, Ruben, Fennelly, Max, Norris, Sean, Wright, Nick, Niblo, Graham, Brodzki, Jacek and Bialek, Janusz (2014) Hierarchical spectral clustering of power grids. IEEE Transactions on Power Systems, 1-9.
Wang, Zhen, Capraro, Valerio, Smyth, Conor, Mylona, Kalliopi and Niblo, Graham A. (2014) Benevolent characteristics promote cooperative behaviour among humans. PLoS ONE, 9, (8), e102881.
Niblo, Graham and Kar, Aditi (2013) A topological splitting theorem for Poincar\'e duality groups and high dimensional manifolds. Geometry & Topology, 17, 2203-2221.
Brodzki, Jacek, Niblo, Graham A., Špakula, Ján, Willett, Rufus and Wright, Nick J. (2013) Uniform local amenability. Journal of Non Commutative Geometry, 7, (2), 583-603.
Brodzki, Jacek, Niblo, Graham A., Nowak, Piotr and Wright, N.J. (2012) A homological characterization of topological amenability. Algebraic & Geometric Topology, 12, (3), Summer Issue, 1767-1780.
Brodzki, Jacek, Niblo, Graham A. and Wright, Nick (2012) Pairings, duality, amenability and bounded cohomology. Journal of the European Mathematical Society, 14, (5), Autumn Issue, 1513-1518.
Kar, Aditi and Niblo, Graham A. (2012) Some non-amenable groups. Publicacions Matemàtiques, 56, (1), Spring Issue, 255-259. (10.5565).
Niblo, Graham and Kar, Aditi (2011) Relative ends, L^2 invariants and Property (T). Journal of Algebra, 333, (1), Spring Issue, 232-240.
Niblo, Graham A. and Guentner, Erik (2011) Complexes and exactness of certain Artin groups. Algebraic and geometric topology, 11, (3), Summer Issue, 1471-1495.
Brodzki, Jacek, Niblo, Graham A., Nowak, Piotr and Wright, Nick (2010) Amenable actions, invariant means and bounded cohomology. Journal of Topology and Analysis, 4, (321), Autumn Issue, 321-334.
Brodzki, Jacek, Niblo, Graham A. and Wright, Nick (2010) A cohomological characterisation of Yu's Property A for metric spaces. Geometry & Topology, 16, 391-432.
Brodzki, J, Niblo, G.A, Putwain, R.J and Wright, N.J (2010) K-theory for subspaces of groups. Pre-print
Brodzki, J., Campbell, S.J., Guentner, E., Niblo, G.A. and Wright, N.J. (2009) Property A and CAT(0) cube complexes. Journal of Functional Analysis, 256, (5), 1408-1431.
Brodzki, Jacek, Niblo, Graham A. and Wright, Nick J. (2009) Partial translation algebras for trees. Journal of Noncommutative Geometry, 3, (1), 83-98.
Niblo, Graham A. and Sageev, Michah (2008) On the Kropholler conjecture. 54
Brodzki, Jacek, Niblo, Graham A. and Wright, Nicholas (2008) Property A and exactness of the uniform Roe algebra. L’Enseignement Mathématique (2), 54, 46-48.
Brodzki, Jacek, Niblo, Graham A. and Wright, Nick (2007) Property A, partial translation structures and uniform embeddings in groups. Journal of the London Mathematical Society
Chatterji, Indira and Niblo, Graham A. (2007) A characterization of hyperbolic spaces. Groups, Geometry and Dynamics, 1, (3), 281-299.
Anderson, James W., Fox, Keith R. and Niblo, Graham A. (2006) A fast algorithm for the construction of universal footprinting templates in DNA. Journal of Mathematical Biology, 52, (3), 307-342.
Niblo, Graham, Sageev, Michah, Scott, Peter and Swarup, Gadde A. (2005) Minimal cubings. International Journal of Algebra and Computation, 15, (2), 343-366.
Campbell, Sarah and Niblo, Graham A. (2005) Hilbert space compression and exactness of discrete groups. Journal of Functional Analysis, 222, (2), 292-305.
Chatterji, Indira and Niblo, Graham (2005) From wall spaces to CAT(0) cube complexes. International Journal of Algebra and Computation, 15, (5-6), 875-885.
Niblo, Graham (2004) A geometric proof of Stallings' Theorem on groups with more than one end. Geometriae Dedicata, 105, (1), 61-76.
Niblo, G.A. and Reeves, L.D. (2003) Coxeter Groups act on CAT(0) cube complexes. Journal of Group Theory, 6, (3), 399-413.
Niblo, G.A. (2002) The singularity obstruction for group splittings. Topology and its Applications, 119, (1), 17-31.
Niblo, G.A. and Williams, B.T. (2002) Engulfing in word-hyperbolic groups. Algebraic and Geometric Topology, 2, 743-755.
Niblo, Graham A. and Wise, Daniel T. (2001) Subgroup separability, knot groups, and graph manifolds. Proceedings of the American Mathematical Society, 129, (3), 685-693.
Niblo, Graham A. (1999) Double coset decompositions of groups. Journal of Algebra, 220, (2), 512-518.
Niblo, Graham A. and Reeves, Lawrence D. (1998) The geometry of cube complexes and the complexity of their fundamental groups. Topology, 37, (3), 621-633.
Niblo, Graham and Reeves, Lawrence (1997) Groups acting on CAT(0) cube complexes. Geometry and Topology, 1, 1-7.
Niblo, Graham A. and Roller, Martin A. (1996) Groups acting on cubes and Kazhdan's property (T). Proceedings of the American Mathematical Society, 126, (3), 693-699.
Niblo, G.A. (1995) Finding splittings of groups and three-manifolds. Bulletin of the London Mathematical Society, 27, 567-574.

## Book Section

Brodzki, Jacek and Niblo, Graham A. (2006) Approximation properties for discrete groups. In, Bojarski, B., Mishchenko, A., Troitsky, E. and Weber, A. (eds.) C*-algebras and Elliptic Theory. Basel, Switzerland, Birkhäuser, 23-35. (Trends in Mathematics).
Leary, I.J., Niblo, G.A. and Wise, D.T. (1999) Some free-by-cyclic groups. In, Campbell, C.M., Robertson, E.F., Ruscuc, N. and Smith, G.C. (eds.) Groups St Andrews 1997 in Bath. Cambridge, UK, Cambridge University Press, 512-516. (London Mathematical Society Lecture Note Series 261). .
Niblo, Graham A. and Wise, Daniel T. (1998) The engulfing property for 3-manifold groups. In, Rivin, Igor, Rourke, Colin and Series, Caroline (eds.) The Epstein Birthday Schrift. Coventry, UK, Geometry & Topology Publications, 413-418. (Geometry & Topology Monographs, 1). .

### Research Interests

My principle research interests lie at the intersection of geometry, topology, analysis and group theory. Recent projects include new homological characterisations of amenability and of property A generalising results of Johnson and Block and Weinberger. These results exposed an unexpected relationship between the classical results for amenability and led to new characterisations of generalisations such as the notion of a topologically amenable action on a compact space, or C* exactness for a group. This work does have a functional analytic flavour, arising from the need to construct invariant means as limits. For example the integers are known to admit an invariant mean but no-one has ever seen one in the wild; they are observed to exist as weak limits of geometrically defined subsets known as Folner sets. There has been a lot of interest recently in the use of ideas from geometric group theory to construct such approximations to a mean and this is an ideal area for further exploration, allowing us to use geometry for constructions of objects discovered via analytic methods.

While much of geometric group theory arose from ideas in low dimensional topology (study of the mapping class group, the JSJ decomposition and Stallings' theorem all played a fundamental role) there is increasing interest in tackling problems in high dimensional topology. Given the remarkable recent proof of the Farrell-Jones conjecture for CAT(0) groups (by Bartels and Lueck), there is likely to be scope for a lot more interaction between these areas. In this direction my recent work with our Postdoc Aditi Kar shows that the classical sphere and torus theorems have a generalisation for even dimensional manifolds of real rank at least 2 and for quaternionic manifolds of dimension at least 8. Broadly speaking we show that any $\pi_1$-injective map of such a manifold to an aspherical manifold of dimension 1 higher is homotopic to a finite cover of an embedding. This may be viewed as a topological analogue of the celebrated geometric superrigidity theorem, at least in even dimensions, and we continue to explore the odd dimensional case.

I am also interested in applications of coarse geometry and graph theory to real world problems. My paper with Jim Anderson on construction of universal footprinting templates in DNA is currently being converted to a practical algorithm by an undergraduate Nuffield scholar with plans for the resulting protein assay templates to be built by our collaborator Keith Fox in the new Life Sciences Institute. More recently I have become involved in a project to model catastrophic failure in the the National Grid with the hope of developing new methods to prevent cascading blackouts. Tools from analytic graph theory are being used to detect optimal network decompositions during a crisis. this is joint work with Jacek Brodzki, Ruben Sanchez, Nick Wright and our student Max Fennelly at Southampton, together with engineers and mathematicians from Durham and Edinburgh.

Research projects
1. Topological super-rigidity - a generalisation of the sphere and torus theorems and the JSJ decomposition to high dimensional topology.
2. Topologically amenable actions - generalisations of amenability to a wide variety of geometric and topological contexts. Funded by EPSRC grant
3. Applications of coarse geometry and analytic graph theory - protective islanding of energy networks, homological methods in data analysis. Funded by EPSRC grant

Primary research group:  Pure Mathematics

### Contact

Professor Graham Niblo
Building 54
Mathematical Sciences
University of Southampton
Highfield
Southampton SO17 1BJ

Room Number: 54/8007

Telephone: (023) 8059 3674
Facsimile: (023) 8059 5147
Email: G.A.Niblo@soton.ac.uk