Memoirs of the American Mathematical Society 2009; 84 pp; softcover Volume: 198 ISBN10: 0821842587 ISBN13: 9780821842584 List Price: US$69 Individual Members: US$41.40 Institutional Members: US$55.20 Order Code: MEMO/198/924
 This work is devoted to the isomorphism problem for split KacMoody 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 KacMoody groups, whose image is bounded. Since KacMoody 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 KacMoody groups over arbitrary fields of cardinality at least \(4\). In particular, he obtains a detailed description of automorphisms of KacMoody groups. This provides a complete understanding of the structure of the automorphism group of KacMoody groups over ground fields of characteristic \(0\). The same arguments allow to treat unitary forms of complex KacMoody 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 nonexistence of cocentral homomorphisms of KacMoody groups of indefinite type over infinite fields with finitedimensional target. This provides a partial solution to the linearity problem for KacMoody groups. Table of Contents  The objects: KacMoody groups, root data and Tits buildings
 Basic tools from geometric group theory
 KacMoody groups and algebraic groups
 Isomorphisms of KacMoody groups: an overview
 Isomorphisms of KacMoody groups in characteristic zero
 Isomorphisms of KacMoody groups in positive characteristic
 Homomorphisms of KacMoody groups to algebraic groups
 Unitary forms of KacMoody groups
 Bibliography
 Index
