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Summary

This introductory chapter provides a first glimpse at the principal characters of the book: Albert algebras and octonions, but also at other members of the families of Freudenthal and composition algebras. They are presented here in the familiar surroundings of the field of real numbers and over the integers. This has the advantage of following rather closely the historical development of the subject (stretching back into the nineteenth century), and of providing a first motivation for the study of quadratic Jordan algebras. Following Zorn (1933), we define the algebra of Graves–Cayley octonions, allowing us to view the Hamiltonian quaternions as an appropriate subalgebra. We describe in detail the Z-algebras of Hurwitz quaternions (1896) and of Dickson–Coxeter octonions (1923, 1946), which in turn give rise to our first encounter with Albert algebras over the integers.

We prove the following: if a group $\Gamma$ is torsion-free, and relatively hyperbolic (with the Bounded Coset Penetration property), relative to a subgroup admitting a finite classifying space, then $\Gamma$ admits a finite classifying space. In this case, if the subgroup admits a boundary in the sense of $\mathcal{Z}$-structures, we prove that $\Gamma$ admits a boundary. This extends classical results of Rips, and of Bestvina and Mess to the relative case.

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I - Prologue: the ancient protagonists

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Classifying Spaces and Boundaries for Relatively Hyperbolic Groups

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2 results

I - Prologue: the ancient protagonists

Summary

Classifying Spaces and Boundaries for Relatively Hyperbolic Groups