This project has two strands: algebraic versus geometric invariants of groups, and exploring non-positively curved groups. Both fall within the orbit of Gromov's study of infinite discrete groups through large-scale geometry, heavily driven by parallels with Riemannian geometry. The first focuses on filling invariants, which concern the geometric features of discs spanning loops ("soap film geometry") in spaces associated with groups. The second concerns how notions of non-positive curvature in group theory interrelate and the wildness present in non?positively curved groups, particularly in their conjugacy problems and subgroups.
The work falls within the subject of Geometric Group Theory, which concerns infinite groups and the spaces on which they act. Its roots lie in the early 20th century, particularly in low-dimensional and algebraic topology. It has taken off since the late 1980s when it became clear that geometric features of the spaces involved have profound repercussions for the algebraic structure of the groups. Rich interactions have ensued with, for example, Riemannian geometry, probability, Lie groups, dynamical systems, ergodic theory, combinatorics, computer science and logic. In addition to the research work, a course in Geometry and Topology will be deigned for undergraduates whose studies who are not primarily concentrated in mathematics.
Intellectual Merit This project was on a topics in Geometric Group Theory, which concerns infinite groups and the spaces on which they act. Its roots lie in the early 20th century, particularly in low-dimensional and algebraic topology. It has taken off since the late 1980s when it became clear that geometric features of the spaces involved have profound repercussions for the algebraic structure of the groups. This project had two outstanding outcomes. The first was the production (with O. Baker) of an example of a "hyperbolic group" and subgroup for which there is no "Cannon–Thurston map". A group is hyperbolic when it displays exclusively negative curvature. Hyperbolic groups (and spaces) have been studied fruitfully via their "boundaries" - that is, by studying the routes off to infinity. When one hyperbolic space sits inside another, one can examine how the boundary of the first sits inside that of each other. This has proved productive and influential in the study of 3-manifolds. However what we show that this breaks down in the greater generality of groups. Baker and the PI also investigated the relationship between Cannon–Thurston maps and subgroup distortion. We proved that Cannon–Thurston maps are well-defined (that is, subgroup inclusion induces a map of the boundaries) for heavily distorted free subgroups inside the family of hyperbolic groups known as hyperbolic hydra. Whilst this indicates that distortion is not an obstacle to the map being well-defined, we show that heavy subgroup distortion always manifests in Cannon–Thurston maps (when they are well–defined) in that their continuity is quantifiably wild. The second outstanding outcome of the project is a polynomial time solution (with W. Dison and E. Einstein) for the "Word Problem" for a "hydra group". Dison and the PI had previously shown that a "hydra phenomenon" gives rise to novel groups with extremely fastgrowing (Ackermannian) "Dehn functions". The Word Problem asks for an algorithm which declares whether or not words on the generators represent theidentity. The Dehn function is a complexity measure of a direct attackon the Word Problem by applying the defining relations. This work required us to design algorithms to compute efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions. Another project outcome was that N. Brady, W. Dison and the PI completed our construction of "hyperbolic hydra": examples of hyperbolic groups with finite-rank free subgroups of huge (Ackermannian) distortion. Also M. Conder and the PI completed an account of a series of lectures of Graham Higman on januarials, namely coset graphs for actions of triangle groups which become 2-face maps when embedded in orientable surfaces. The PI wrote a chapter for a forthcoming book "Office hours with a geometric group theorist" edited by Dan Margalit and Matt Clay. And A. Sale and the PI proved a number of results on groups with "finite palindromic width" - that is, groups for which there exists n such that every element can be expressed as a product of n or fewer palindromic words. Also, the PI and his graduate student Amchislavska worte an article a family of metabelian groups generalizing lamplighters which have beautiful geometric realizations as horocyclic products of trees. Broader impact The work on computing efficiently with enormous integers, represented in compressed forms by strings of Ackermann functions described above, should be of significance for computer science as it compares with the study of power circuits and straight-line programs. The PI trained three graduate students, Margarita Amchislavska (graduated July 2014), Yash Lodha (graduation expected 2014–15), and Kristen Pueschel (graduation expected summer 2016) during this project. Additionally, the PI worked with young researchers Eduard Einstein (a Cornell graduate student), Owen Baker, and Andrew Sale, and contributed to guiding Baker and Sale into good postdoc. positions. The PI was a main organizer for the Cornell Topology Festival in 2011 and 2012. The 2012 Festival was an expanded meeting celebrating the fiftieth anniversary; we had 222 partipants and an exceptional line-up of speakers. The PI was also an orgnizer of a workshop on cubical complexes and their applications at the International Centre for Mathematical Sciences in Edinburgh in 2012.
