28,014 research outputs found
Automorphism groups of polycyclic-by-finite groups and arithmetic groups
We show that the outer automorphism group of a polycyclic-by-finite group is
an arithmetic group. This result follows from a detailed structural analysis of
the automorphism groups of such groups. We use an extended version of the
theory of the algebraic hull functor initiated by Mostow. We thus make
applicable refined methods from the theory of algebraic and arithmetic groups.
We also construct examples of polycyclic-by-finite groups which have an
automorphism group which does not contain an arithmetic group of finite index.
Finally we discuss applications of our results to the groups of homotopy
self-equivalences of K(\Gamma, 1)-spaces and obtain an extension of
arithmeticity results of Sullivan in rational homotopy theory
Weakly commensurable groups, with applications to differential geometry
The article contains a survey of our results on weakly commensurable
arithmetic and general Zariski-dense subgroups, length-commensurable and
isospectral locally symmetric spaces and of related problems in the theory of
semi-simple agebraic groups. We have included a discussion of very recent
results and conjectures on absolutely almost simple algebraic groups having the
same maximal tori and finite-dimensional division algebras having the same
maximal subfields.Comment: Improved exposition, updated bibliography. arXiv admin note:
substantial text overlap with arXiv:1212.121
Cusps of arithmetic orbifolds
This thesis investigates cusp cross-sections of arithmetic real, complex, and
quaternionic hyperbolic --orbifolds. We give a smooth classification of
these submanifolds and analyze their induced geometry. One of the primary tools
is a new subgroup separability result for general arithmetic lattices.Comment: 76 pages; Ph.D. thesi
Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces
We introduce the notion of weak commensurabilty of arithmetic subgroups and
relate it to the length equivalence and isospectrality of locally symmetric
spaces. We prove many strong consequences of weak commensurabilty and derive
from these many interesting results about isolength and isospectral locally
symmetric spaces.Comment: 62 page
Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces
The article contains a survey of results on length-commensurable and
isospectral locally symmetric spaces and related problems in the theory of
semi-simple algebraic groups.Comment: New material has been added in section
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