Chancellor’s Scholarship Application 2015-16 Damian Sercombe

PhD Research Proposal Covolumes of cocompact lattices in automorphism groups of Davis complexes Prospective Supervisors – Professor Brian Bowditch and Dr Saul Schleimer This research proposal is in , a field which uses geometric techniques to study the algebraic structure of groups. One is guided by Gromov’s (1987) program of classifying all finitely generated groups up to quasi-isometry [5]. Some background is as follows. A topological group is a group that is endowed with a compatible topological structure. A locally compact group is a topological group that is locally compact as a topological space. It is known that a locally compact group naturally admits an important measure called the Haar measure [11]. A lattice in a locally compact group is a discrete subgroup so that the quotient admits a finite invariant measure (known as a covolume). If this quotient is compact, then the lattice is called cocompact. A basic question is: what is the set of possible (positive real) values that may be taken by the covolumes of lattices of such a group [10, 11, 12]? This question was first studied in 1945 by Siegel [9] for the case of SL(2,ℝ). It has since been pursued by various researchers for the case of a semi-simple algebraic group over a field. More recently, this question has been studied by Bass & Lubotzky [1] and Burger & Mozes [4] for the automorphism group of a tree and a product of trees, respectively this research proposal is concerned with the automorphism groups of a naturally occurring class of polyhedral complexes (called Davis complexes) that are associated with Coxeter groups [11]. Now specialise to the case where X is a Davis complex and G is its (locally compact) automorphism group. It is known that a subgroup of G is a cocompact lattice precisely when it acts on X with finite stabilisers and a compact quotient [11]. Thomas showed that the set S of covolumes of lattices (not necessarily cocompact!) in G is nondiscrete, by explicitly constructing a family of lattices using a tool called group actions on complexes of groups [11]. White showed that if X is a tree then this set S has no positive lower bound [12]. Finally, in 2014 (under the supervision of Thomas), the author constructed an original family of graphs of groups and used Bass-Serre theory to show that if X is a tree, then the subset of S for cocompact lattices is non-discrete [7]. It remains an open problem – suggested by Thomas as suitable in scale and depth for a PhD project – to completely determine the set of covolumes of cocompact lattices in the automorphism group of an arbitrary Davis complex. Some proposed methods are as follows.  To begin with the simplest case where the Davis complex is a tree [1, 12].  One may restrict the set of all possible covolumes by using a combinatorial result of Thomas’ which prohibits certain covolumes for a given polyhedral complex [10].  A generalisation of Bass-Serre theory, called complexes of groups, can be used to construct a cocompact lattice in the automorphism group of a Davis complex X as the fundamental group of a complex of groups Y with universal cover X (not to be confused with their topological analogues) [7, 11]. Then a result of Serre’s [8] gives an explicit formula for the covolume as the sum of reciprocals of the orders of the local groups in Y.  Alternatively, one could follow White [12] and construct a family of automorphisms in a Coxeter group which induces a family of automorphisms in the associated Davis complex. That is, one could explicitly construct a group that acts faithfully on a Davis complex with finite stabilisers and a compact quotient. Chancellor’s Scholarship Application 2015-16 Damian Sercombe

Broader Impact – We state here two concrete applications of geometric group theory to the theory of complex networks, which are ubiquitous in society and include search engines, financial networks, social networks, the internet as well as biological and transportation networks. Recent work in complexity theory has argued that there exists a “hidden” underlying complex networks – in particular the internet – that exhibits negative curvature (that is, there exists a structure preserving map from a complex network to a hyperbolic space) [2, 3]. Research on the geometric properties of certain hyperbolic spaces should therefore aid our understanding of the relationship between the topological structure of complex networks and their function [2]. In particular, such research could help to develop and analyse the consequences of a proposed upgrade to the routing infrastructure of the internet based on a hyperbolic mapping (which is designed to solve current problems with internet scalability) [2, 3]. In addition, geometric group theory has applications to economics and computer science, stemming from the fact that one may endow the set of symmetries of any graph or network with a locally compact group structure. This connection has been used (including by the author) to study topological properties of financial and social networks such as robustness, contagion and dynamics [6, 13].

References [1] Bass, H. and Lubotzky, A., “Tree lattices,” Birkhäuser Boston (2001). [2] Boguna, M., Krioukov, D. and Papadopoulos, F., "Sustaining the Internet with Hyperbolic Mapping," Nature Communications, v.1 (2010): p. 62. [3] Boguna, M., Krioukov, D., Papadopoulos, F., Kitsak, M. and Vahdat, A., " of Complex Networks," Physical Review E, v.82 (2010): p. 036106. [4] Burger, M. and Mozes, S., "Lattices in product of trees," Publications Mathématiques de l'IHÉS 92, no. 1 (2000): pp 151-194. [5] Gromov, M., "Infinite groups as geometric objects," In Proceedings of the International Congress of Mathematicians, vol. 1, p.2. (1984). [6] Newton, J. and Sercombe, D., “Upward bias for rooted spanning trees on a network”, in preparation (2014). [7] Sercombe, D., “A family of uniform lattices acting on a Davis complex with a non- discrete set of covolumes”, in preparation (2014). [8] Serre, J.P., "Trees,” Springer-Verlag, Berlin (2003), ch. 1-5. [9] Siegel, C. L., "Some remarks on discontinuous groups," The Annals of Mathematics 46, no. 4 (1945): pp 708-718. [10] Thomas, A., "Covolumes of uniform lattices acting on polyhedral complexes," Bulletin of the London Mathematical Society 39, no. 1 (2007): pp 103-111. [11] Thomas, A., “Existence, covolumes and infinite generation of lattices for Davis complexes,” Groups Geom. Dyn., 6, no. 4 (2012): pp 765–801. [12] White, G., “Automorphisms of geometric structures associated to Coxeter groups”, preprint, arXiv:1202.6441 (2012). [13] Young, H. P., "The dynamics of social innovation," Proceedings of the National Academy of Sciences 108, no. Supplement 4 (2011): 21285-21291.