Exponential Higher Dimensional Isoperimetric Inequalities for Some Arithmetic Groups

We show that arithmetic subgroups of semisimple groups of relative Q-type An, Bn, Cn, Dn, E6, or E7 have an exponential lower bound to their isoperimetric inequality in the dimension that is 1 less than the real rank of the semisimple group. Let G be a connected, semisimple, Q-group that is almost simple over Q. Let X be the symmetric space of noncompact type associated with G(R) and let XZ be a contractible subspace of X that is a finite Hausdorff distance from some G(Z)-orbit in X; Raghunathan proved that such a space exists [Ra 1]. We denote the R-rank of G by rkRG. Given a homology n-cycle Y⊆XZ we let vX(Y ) be the infimum of the volumes of all (n + 1)-chains B⊆X such that ∂B = Y . Similarly, we let vZ(Y ) be the infimum of the volumes of all (n+1)-chains B⊆XZ such that ∂B = Y . We define the ratio Rn(Y ) = vZ(Y ) vX(Y ) and we let Rn(G(Z)) : R>0 → R≥1 be the function Rn(G(Z))(L) = sup{Rn(Y ) | vol(Y ) ≤ L } These functions measure a contrast between the geometries of G(Z) and X. Clearly if G is Q-anisotropic (or equivalently, if G(Z) is cocompact in G(R)) then we may take XZ = X so that Rn(G(Z)) = 1 for all n. The case is different when G is Q-isotropic, or equivalently, if G(Z) is non-cocompact in G(R). Leuzinger-Pittet conjectured that RrkRG−1(G(Z)) is bounded below by an exponential when G is Q-isotropic [L-P]. The conjecture in the case rkRG = 1 is equivalent to the well-known observation that the word metric for non-cocompact lattices in rank one real simple Lie groups is exponentially distorted in its corresponding symmetric space. Prior to [L-P], the conjecture was evidenced by other authors in some cases. It was proved by Epstein-Thurston when G(Z) = SLk(Z) 1