In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over \(\mathbb{Q}\) and its complex Langlands' dual. The authors give a new proof of the “Saturation Conjecture” for \(GL(\ell)\) as a consequence of their solution of the corresponding “saturation problem” for the Hecke structure constants for all split reductive algebraic groups over \(\mathbb{Q}\).
The key idea in geometric group theory is to
study infinite groups by endowing them with a metric and treating them
as geometric spaces. This applies to many groups naturally appearing
in topology, geometry, and algebra, such as fundamental groups of
manifolds, groups of matrices with integer coefficients, etc. The
primary focus of geometric group theory is to cover the foundations of
geometric group theory, including coarse topology, ultralimits and
asymptotic cones, hyperbolic groups, isoperimetric inequalities,
growth of groups, amenability, Kazhdan's Property (T) and the Haagerup
property, as well as their characterizations in terms of group actions
on median spaces and spaces with walls.
The book contains proofs of several fundamental results of
geometric group theory, such as Gromov's theorem on groups of
polynomial growth, Tits's alternative, Stallings's theorem on ends of
groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem,
and quasiisometric rigidity theorems of Tukia and Schwartz. This is
the first book in which geometric group theory is presented in a form
accessible to advanced graduate students and young research
mathematicians. It fills a big gap in the literature and will be used
by researchers in geometric group theory and its applications.
Graduate students and researchers interested in geometric group theory.