Lattices of minimal covolume in SLn(R)

Abstract

The objective of this paper is to determine the lattices of minimal covolume in prefixSLnfalse(double-struckRfalse), for n⩾3. The answer turns out to be the simplest one: prefixSLnfalse(double-struckZfalse) is, up to automorphism, the unique lattice of minimal covolume in prefixSLnfalse(double-struckRfalse). In particular, lattices of minimal covolume in prefixSLnfalse(double-struckRfalse) are non‐uniform when n⩾3, contrasting with Siegel's result for prefixSL2false(double-struckRfalse). This answers for prefixSLnfalse(double-struckRfalse) the question of Lubotzky: is a lattice of minimal covolume typically uniform or not?

Quaternionic hyperbolic lattices of minimal covolume

“…A more concrete description in those cases is only available in low dimensions (in the form of Coxeter groups) or in a few special cases (see, for instance, [8,9]). Another situation where satisfactory descriptions are available is the case of a split Lie group G (see, for instance, [35] and [18,31] in the positive characteristic case).…”