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"Abstract" Homomorphisms of Split Kac-Moody Groups
Pierre-Emmanuel Caprace, Université Libre de Bruxelles, Belgium
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Memoirs of the American Mathematical Society
2009; 84 pp; softcover
Volume: 198
ISBN-10: 0-8218-4258-7
ISBN-13: 978-0-8218-4258-4
List Price: US$65 Individual Members: US$39
Institutional Members: US\$52
Order Code: MEMO/198/924

This work is devoted to the isomorphism problem for split Kac-Moody groups over arbitrary fields. This problem turns out to be a special case of a more general problem, which consists in determining homomorphisms of isotropic semisimple algebraic groups to Kac-Moody groups, whose image is bounded. Since Kac-Moody groups possess natural actions on twin buildings, and since their bounded subgroups can be characterized by fixed point properties for these actions, the latter is actually a rigidity problem for algebraic group actions on twin buildings. The author establishes some partial rigidity results, which we use to prove an isomorphism theorem for Kac-Moody groups over arbitrary fields of cardinality at least $$4$$. In particular, he obtains a detailed description of automorphisms of Kac-Moody groups. This provides a complete understanding of the structure of the automorphism group of Kac-Moody groups over ground fields of characteristic $$0$$.

The same arguments allow to treat unitary forms of complex Kac-Moody groups. In particular, the author shows that the Hausdorff topology that these groups carry is an invariant of the abstract group structure.

Finally, the author proves the non-existence of cocentral homomorphisms of Kac-Moody groups of indefinite type over infinite fields with finite-dimensional target. This provides a partial solution to the linearity problem for Kac-Moody groups.