Exponential higher dimensional isoperimetric inequalities for some arithmetic groups

TL;DR

This paper establishes exponential lower bounds for isoperimetric inequalities in certain arithmetic groups, revealing new geometric properties related to their algebraic structure.

Contribution

It provides the first exponential lower bounds for isoperimetric inequalities in arithmetic subgroups of specific semisimple groups, advancing understanding of their geometric complexity.

Findings

Arithmetic subgroups of specified semisimple groups have exponential isoperimetric bounds.

The results apply to groups of relative Q-type A_n, B_n, C_n, D_n, E_6, E_7.

The bounds are in the dimension one less than the real rank of the groups.

Abstract

We show that arithmetic subgroups of semisimple groups of relative Q-type A_n, B_n, C_n, D_n, E_6, or E_7 have an exponential lower bound to their isoperimetric inequality in the dimension that is 1 less than the real rank of the semisimple group.