Volume 2015, Issue 704

We show that the Halphen transform of a Lamé equation can be written as the symmetric square of the Lamé equation followed by an Euler transform. We use this to compute a list of Lamé equations with arithmetic Fuchsian monodromy group. It contains all those Lamé equations where the quaternion algebra A associated to the arithmetic Fuchsian group is defined over ℚ.

We define an extended Bloch group for an arbitrary field F , and show that this group is naturally isomorphic to K3ind(F)$K_3^{\operatorname{ind}}(F)$ if F is a number field. This gives an explicit description of K3ind(F)$K_3^{\operatorname{ind}}(F)$ in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3-manifold with finite volume and invariant trace field k has a fundamental class [M]$[M]$ in K3ind(k)⊗ℤ[12]$K_3^{\operatorname{ind}}(k)\otimes \mathbb {Z}[\frac{1}{2}]$. If M is closed, 2[M]$2[M]$ is well defined in K3ind(k)$K_3^{\operatorname{ind}}(k)$, and if M has cusps, 8[M]$8[M]$ is well defined in K3ind(k)$K_3^{\operatorname{ind}}(k)$.

We discuss various types of Tits alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the theory of parabolic subgroups, which allows to describe the structure of subgroups of graph products that contain no non-abelian free subgroups. We also obtain a number of results regarding the stability of some group properties under taking graph products.

In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in an Apollonian sphere packing.

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in ∂ M . Our main result is that for any g ≥ 0, the space of simple closed curves in ∂ M where all the absolutely area minimizing surfaces they bound in M has genus ≥ g is open and dense in the space 𝒜 of nullhomologous simple closed curves in ∂ M . For showing this we prove a bridge principle for absolutely area minimizing surfaces. Moreover, we show that for any g ≥ 0, there exists a curve γ g in 𝒜 such that the minimum genus of the absolutely area minimizing surfaces γ g bounds is exactly g . As an application of these results, we further prove that the simple closed curves in ∂ M bounding more than one minimal surface in M is an open and dense subset of 𝒜. We also show that there are disjoint simple closed curves in ∂ M bounding minimal surfaces in M which are not disjoint. This allows us to answer a question of Meeks, by showing that for any strictly mean convex 3-manifold M , there exists a simple closed curve Γ in ∂ M which bounds a stable minimal surface which is not embedded. We also gave some applications of these results to the simple closed curves in ℝ 3 .

We use the methods of topological Hochschild homology to shed new light on groups satisfying the Bass trace conjecture. Factorization of the Hattori–Stallings rank map through the Bökstedt–Hsiang–Madsen cyclotomic trace map leads to Linnell's restriction on such groups. As a new consequence of this restriction, we show that the conjecture holds for any group G wherein every subgroup isomorphic to the additive group of rational numbers has nontrivial and central image in some quotient of G .

Soit F une extension quadratique de ℚ p . Nous prouvons que les représentations lisses irréductibles supersingulières à caractère central de GL 2 ( F ) ne sont pas de présentation finie. Let F be a quadratic extension of ℚ p . We prove that smooth irreducible supersingular representations with central character of GL 2 ( F ) are not of finite presentation.

We introduce the mean curvature flow of curves in the Minkowski plane ℝ 1,1 and give a classification of all the self-similar solutions. In addition, we describe five other exact solutions to the flow.