He introduced the theory of
buildings (sometimes known as
Tits buildings), which are combinatorial structures on which groups act, particularly in
algebraic group theory (including
finite groups, and groups defined over the
p-adic numbers). The related theory of
(B, N) pairs is a basic tool in the theory of
groups of Lie type. Of particular importance is his classification of all irreducible buildings of spherical type and rank at least three, which involved classifying all
polar spaces of rank at least three. The existence of these buildings initially depended on the existence of a group of Lie type in each case, but in joint work with
Mark Ronan he constructed those of rank at least four independently, yielding the groups directly. In the rank-2 case spherical building are
generalized n-gons, and in joint work with Richard Weiss he classified these when they admit a suitable group of symmetries (the so-called
Moufang polygons). In collaboration with
François Bruhat he developed the theory of affine buildings, and later he classified all irreducible buildings of affine type and rank at least four. Another of his well-known theorems is the "
Tits alternative": if
G is a
finitely generated subgroup of a
linear group, then either
G has a
solvable subgroup of
finite index or it has a
free subgroup of rank 2. The
Tits group and the
Kantor–Koecher–Tits construction are named after him. He introduced the
Kneser–Tits conjecture. == Publications ==