We show that there is a canonical, order preserving map ψ of lattices of subgroups, which maps the lattice Sub(A) of subgroups of the ideal class group of a galois number field K into the lattice Sub(H/K) of subfields of the Hilbert class field. Furthermore, this map is a capitulation map in the sense that all the primes in the classes of A ⊂ A capitulate in ψ(A). In particular we have a new, s...

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, have positive first Betti number, a homomorphisms onto non-Abelian free group. For we determine minimal neat subgroup. Finally, answer question raised by Stover describe an infinite tower ball quotients all with single cusp.

We study invariant random subgroups (IRSs) of semidirect products G = AoΓ. In particular, we characterize all IRSs of parabolic subgroups of SLd(R), and show that all ergodic IRSs of Rdo SLd(R) are either of the form RdoK for some IRS of SLd(R), or are induced from IRSs of Λo SL(Λ), where Λ < Rd is a lattice.

Based on the ideas in some recently uncovered notes of Selberg [14] on discrete subgroups of a product of SL2(R)’s, we show that a discrete subgroup of SL3(R) generated by lattices in upper and lower triangular subgroups is an arithmetic subgroup and hence a lattice in SL3(R).

The Leech lattice is a remarkable 24 dimensional Euclidean lattice with more than 8 10 18 symmetries. Suppose one is given two vectors in. Are they equivalent under one of these symmetries? We use the geometry of the lattice to produce an algorithm to answer this question quickly. Along the way we produce fast algorithms to detect whether two vectors in R 24 are equivalent under the action of o...

Example 1. Let G be a finite group. Then the set of all subgroups of G, partially ordered by inclusion, is a lattice. For H ≤ G the sublattice OG(H) of overgroups of H in G is the interval sublattice [H,G]. Call such a lattice a finite group interval lattice. There is a well-known open question as to whether every nonempty finite lattice is isomorphic to a finite group interval lattice. See [PP...