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Natural sciences
- Group theory and generalisations
- Topological groups, Lie groups
- Geometry
In 1974, Tits published his seminal Springer Lecture Notes in which he classified spherical buildings of rank at least 3, or, equivalently, spherical BN-pairs of rank at least 3. In his book, Tits noted that infinite BN-pairs of rank 2 could not be classified without extra assumptions, and conjectured how the finite BN-pairs of rank 2 would look like. In the meanwhile, this conjecture has been confirmed by several authors, but not without entailing the Classification of Finite Simple Groups (CFSG) — a result which encompasses hundreds of papers totaling tens of thousands of pages.
Tits had a classification-free proof in mind though — a geometrical proof.
In this proposal, we will develop new foundations in the theory of generalized polygons (which admit an automorphism group with a BN-pair), so as to solve Tits’s problem as a consequence. We will also consider far-reaching generalizations of Tits’s problem, but do allow the use of CFSG in order to reach the latter goals.