  We investigate several problems in relativity and particle physics where
symmetries play a central role; in all cases geometric properties of Lie groups
and their quotients are related to physical effects. The first part is
concerned with symmetries in gravity. We apply the theory of Lie group
deformations to isometry groups of exact solutions in general relativity,
relating the algebraic properties of these groups to physical properties of the
spacetimes. We then make group deformation local, generalising deformed special
relativity (DSR) by describing gravity as a gauge theory of the de Sitter
group. We find that in our construction Minkowski space has a connection with
torsion; physical effects of torsion seem to rule out the proposed framework as
a viable theory. A third chapter discusses a formulation of gravity as a
topological BF theory with added linear constraints that reduce the symmetries
of the topological theory to those of general relativity. We discretise our
constructions and compare to a similar construction by Plebanski which uses
quadratic constraints. In the second part we study CP violation in the
electroweak sector of the standard model and certain extensions of it. We
quantify fine-tuning in the observed magnitude of CP violation by determining a
natural measure on the space of CKM matrices, a double quotient of SU(3),
introducing different possible choices and comparing their predictions for CP
violation. While one generically faces a fine-tuning problem, in the standard
model the problem is removed by a measure that incorporates the observed quark
masses, which suggests a close relation between a mass hierarchy and
suppression of CP violation. Going beyond the standard model by adding a
left-right symmetry spoils the result, leaving us to conclude that such
additional symmetries appear less natural.
