IV.10. Geometric and Combinatorial Group Theory 431 program, the 1979 American Mathematical Society vol- ume titled Automorphic Forms, Representations, and L-functions (but universally known as "The Corvallis Proceedings") is more advanced, and as good a place to start as any. in L G( C ). 15 However, these are not conjugacy classes of elements of L G( C ), as before, but of homomorphisms from the Galois group of k to L G. The Langlands dual was originally defined in a combinatorial manner, but there is now a conceptual definition. A few examples of pairs (G, L G) are (GL n , GL n ), (SO 2n+1 , Sp 2n ), and (SL n , PGL n ). In this way the Langlands program describes the rep- resentation theory as built out of the structure of G and the arithmetic of k. Although this description indicates the flavor of the conjectures, it is not quite correct as stated. For instance, one has to modify the Galois group 16 in such a way that the correspondence is true for the group GL 1 (k) = k . When k = R , we get the representation theory of R (or its compact form S 1 ), which is Fourier analysis; on the other hand, when k is a p-adic local IV.10 Geometric and Combinatorial Group Theory Martin R. Bridson 1 What Are Combinatorial and Geometric Group Theory? Groups and geometry are ubiquitous in mathematics, groups because the symmetries (or automorphisms [I.3 §4.1]) of any mathematical object in any context form a group and geometry because it allows one to