A variety of structural transitions in nature do not involve a group-subgroup relationship. A two-dimensional example is the transition between a square lattice and a triangular lattice, which is known to occur in vortex lattices, Wigner crystals, skyrmion lattices, colloids, diblock copolymers, etc. A three-dimensional analog is the shock-induced BCC to HCP transition in iron. We develop a sys...

We enumerate nonisomorphic lattice-square designs yielded by a conventional construction. Constructed designs are speci3ed by words composed from 3nite-3eld elements. These words are permuted by the isomorphism group in question. The latter group contains a direct-product subgroup, acting, respectively, upon the positions and identities of the 3nite-3eld elements. We review enumeration theory f...

A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple sufficient condition that implies G is a duality group. The universal covers of ...

Suppose G is a hyperbolic group whose boundary ∂∞G has topological dimension k. If ∂∞G is quasi-symmetrically homeomorphic to an Ahlfors kregular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group Isom(Hk+1) of hyperbolic (k+1)-space.

Let K be an abelian extension of the rationals. Let S(K) be the Schur group of K and let CC(K) be the subgroup of S(K) generated by classes containing cyclic cyclotomic algebras. We characterize when CC(K) has finite index in S(K) in terms of the relative position of K in the lattice of cyclotomic extensions of the rationals.

We present a topological proof of the following theorem of Benoist-Quint: for a finitely generated non-elementary discrete subgroup Γ1 of PSL(2,R) with no parabolics, and for a cocompact lattice Γ2 of PSL(2,R), any Γ1 orbit on Γ2\PSL(2,R) is either finite or dense.

We show that the subgroup lattice of any finite group satisfies Frankl’s UnionClosed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest.

With any integral lattice Λ in n-dimensional euclidean space we associate an elementary abelian 2-group I(Λ) whose elements represent parts of the dual lattice that are similar to Λ. There are corresponding involutions on modular forms for which the theta series of Λ is an eigenform; previous work has focused on this connection. In the present paper I(Λ) is considered as a quotient of some fini...

Hotzel asks this for semigroups and right congruences; his and the above version are equivalent. Left congruences on M are equivalence relations closed under all left translations; these are thus equivalent to congruences on the free left M -set on one generator. When M is a group, any left congruence is the decomposition of M into the left cosets of some subgroup, so the lattice of left congru...

This survey is intended as a brief introduction to the theory of hyperbolic buildings and their lattices. Hyperbolic buildings are negatively curved geometric objects which also have a rich algebraic and combinatorial structure, and the study of these buildings and the lattices in their automorphism groups involves a fascinating mixture of techniques from many different areas of mathematics. Ro...