Geometric And Functional Analysis 1016-443X/96/0300489-2351.50+0.20/0 Vol. 6, No. 3 (1996) © 1996 Birkh/~user Verlag, Basel ISOPERIMETRIC INEQUALITIES FOR LATTICES IN SEMISIMPLE LIE GROUPS OF RANK 2 E. LEUZINGER AND C. PITTET Abstract Let G be a connected semisimple Lie group with finite center. Let F be an irreducible non-uniform lattice in G. We show that if the real rank of G is 2, then the Dehn (or filling) function of F is exponential. 1. Introduction and Statement of the Theorem Like growth functions of finitely generated groups [Gr], Dehn functions of finitely presented groups yield quasi-isometric invariants [A], [G2], [Gro2]. Dehn functions can be defined in various ways. 1. Topological definition [HaVo]. Let K be a 2-dimensional cell complex. Let c be an edge path in K homotopic to a point. Its length lct is the number of times it passes over 1-cells and its area A(c) is the minimum number of times it passes over 2-cells during any null-homotopy. Among paths c of length less than or equal to n, consider those with maximal area. The Dehn function of K is 8K(n) = max A(c) . 2. Combinatorial definition [G2]. Let (SIR} be a finite presentation of a group F. Let w C F(S) be a word of the free group generated by S and assume that w is trivial in F. That is N i=1 where the relations ri belong to the (symmetric) set R and where us E F(S). Let A(w) be the minimum value of N in such expressions. Let Iw] be the number of letters of w. The Dehn function of (SIR) is ~(n) = max A(w) . t~,l<_n 3. Geometric definition [Gro2], [Pil]. Let X be a Riemannian manifold and let c : S 1 --+ X be a piecewise smooth loop of length /(c) which is homotopic to a point. Let A(c) be the infimum of the areas of (al- most everywhere differentiable) disks with c as boundary. The Dehn (or