Details of Grant 

EPSRC Reference:	EP/I003843/1
Title:	The Geometry and Logic of Groups
Principal Investigator:	Wilton, Dr HJR
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	UCL
Scheme:	Career Acceleration Fellowship
Starts:	31 March 2011	Ends:	31 August 2014	Value (£):	526,766
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 09 Jun 2010 EPSRC Fellowships 2010 Interview Panel B Announced
Summary on Grant Application Form
As a geometric group theorist, I study the interaction between geometry and symmetry. Symmetry is an important concept throughout science, particularly in physics and chemistry, as well as in every area of mathematics. My work touches on many areas of mathematics, including algebra, logic and topology. You can think of topology as ''rubber geometry' - topologists are allowed to stretch objects, but never tear them. I solve algebraic problems by translating them into geometry or topology - I am often amazed at the elegant and simple solutions that arise from this approach to hard algebraic problems.In geometric group theory, we study ''groups via their ''actions on geometric objects: a ''group is the technical term in mathematics for a collection of symmetries of an object; we say that the group ''acts on the object. Mathematicians work with many different sorts of geometries. I am most interested in ''hyperbolic geometry , in which the angles in a triangle add up to less than 180 degrees (unlike the standard ''Euclidean geometry, in which they add up to exactly 180 degrees). Hyperbolic geometry is tricky to visualize, but many different spaces exhibit some form of hyperbolic geometry. For instance, it is a remarkable fact that a ''randomly chosen group will almost certainly act on a hyperbolic space. Such groups are called ''hyperbolic groups .In this project, I will pursue two questions about infinite hyperbolic groups. An infinite group can be a daunting object to study, and both my lines of enquiry involve simplifying one's perspective.My first question concerns ''separability , in which we ask how much of the group we can see if we restrict our attention to actions on finite objects. I propose to prove that certain important kinds of hyperbolic groups are ''subgroup separable . Roughly speaking, this says that, even though the groups in question are infinite, we can recapture a lot of their structure by looking only at their actions on finite objects.This will resolve several important open questions in low-dimensional topology, including a longstanding problem posed by Andrew Casson. The theory of 3-dimensional topology is dominated by several outstanding conjectures; this result will clarify the relationship between them, by showing that two of them are equivalent. It will also provide important evidence for one of these conjectures.My second question concerns those properties of a group that we can describe using only simple logical terms. The ''elementary properties of a group are the ones that you can write down using only ''and , ''or and ''not , as well as ''for all and ''there exists . Taken together, all the elementary properties of a group collectively form the ''elementary theory of that group. I propose to prove, with Daniel Groves, that the elementary theory of every hyperbolic group is ''decidable . This means that every hyperbolic group has an algorithm that takes an elementary property as input and determines whether or not it is true in that group.This project will broaden and clarify recent monumental work on the elementary theory of hyperbolic groups. My work on this subject has already produced theorems of widespread interest, and I expect many more. We believe that a successful resolution of this question will lead to important advances in our understanding of the structure of hyperbolic groups.
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