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Alexander D. Rahm

Publications and source records attributed to Alexander D. Rahm.

11 recordsLinked to original sources
arXiv2025

On the Calegari-Venkatesh conjecture connecting modular forms spaces and algebraic K-Theory

Calegari and Venkatesh did construct, modulo small torsion, a surjection from the degree 2 homology of the rank 2 projective general linear group over a ring of algebraic integers (of odd class number, and with enough embeddings) to the 2nd algebraic K-group of that ring. They asked whether this surjection becomes an isomorphism when passing to the quotient modulo the Eisenstein ideal on the left hand side. We provide a new method (together with numerical examples) to lift elements in the opposite direction, enabled by a theorem in a more general setting, where we exploit a connection between the algebraic K-groups and the Steinberg homology groups.

math.KT↗
arXiv2025

Computations regarding the torsion homology of Oeljeklaus-Toma manifolds

This article investigates the torsion homology behaviour in towers of Oeljeklaus-Toma (OT) manifolds. This adapts an idea of Silver and Williams from knot theory to OT-manifolds and extends it to higher degree homology groups. In the case of surfaces, i.e. Inoue surfaces of type $S^0$, the torsion grows exponentially in both $H_1$ (as was established by Braunling) and $H_2$ (our result) according to a parameter which already plays a role in Inoue's classical paper, and we obtain that the torsion vanishes in all higher degrees. This motivates our presented machine calculations for OT-manifolds of higher dimension.

math.DG↗
arXiv2022

On Farrell-Tate cohomology of GL(3) over rings of quadratic integers

The goal of the present paper is to push forward the frontiers of computations on Farrell-Tate cohomology for arithmetic groups. The conjugacy classification of cyclic subgroups is reduced to the classification of modules of group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell-Tate cohomology computations, we discuss in detail the homological torsion in PGL(3) over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.

math.KT↗
arXiv2018

Simulation study of the electrical tunneling network conductivity of suspensions of hard spherocylinders

Using Monte Carlo simulations, we investigate the electrical conductivity of networks of hard rods as a function of the volume fraction for two tunneling conductance models. For a simple, orientationally independent tunneling model, we observe non-monotonic behaviour of the bulk conductivity as a function of volume fraction at the isotropic-nematic transition. However, this effect is lost if one allows for anisotropic tunneling. We also compute the mesh number of the Kirchhoff network, which turns out to be a simple alternative to the computationally expensive conductivity of large systems in order to get a qualitative estimate.

cond-mat.soft↗
arXiv2013

Complexifiable characteristic classes

We examine the topological characteristic cohomology classes of complexified vector bundles. In particular, all the classes coming from the real vector bundles underlying the complexification are determined.

math.KT↗
arXiv2013

Accessing the cohomology of discrete groups above their virtual cohomological dimension

We introduce a method to explicitly determine the Farrell-Tate cohomology of discrete groups. We apply this method to the Coxeter triangle and tetrahedral groups as well as to the Bianchi groups, i.e. PSL_2 over the ring of integers in an imaginary quadratic number field, and to their finite index subgroups. We show that the Farrell-Tate cohomology of the Bianchi groups is completely determined by the numbers of conjugacy classes of finite subgroups. In fact, our access to Farrell-Tate cohomology allows us to detach the information about it from geometric models for the Bianchi groups and to express it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups in the Bianchi groups have been determined in a thesis of Krämer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically in the appendix. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.

math.KT↗
arXiv2013

On Level One Cuspidal Bianchi Modular Forms

In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms.

math.NT↗
arXiv2012

On a question of Serre

Consider the Borel-Serre compactification of the quotient of hyperbolic 3-space by a finite index subgroup in a Bianchi group, and in particular the following question which Serre posed on page 514 of the quoted article. Consider the map alpha induced on homology when attaching the boundary into the Borel-Serre compactification. How can one determine the kernel of alpha (in degree 1)? Serre used a global topological argument and obtained the rank of the kernel of alpha. In the quoted article, Serre did add the question what submodule precisely this kernel is. Through a local topological study, we can decompose the kernel of alpha into its parts associated to each cusp.

math.KT↗
arXiv2012

Higher torsion in the Abelianization of the full Bianchi groups

Consider the Bianchi groups, namely the SL_2 groups over rings of imaginary quadratic integers. In the literature, there has been so far no example of p-torsion in the integral homology of the full Bianchi groups, for p a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance p = 80737 at the discriminant -1747.

math.KT↗
arXiv2011

Homology and K-theory of the Bianchi groups

We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We use it to explicitly compute their integral group homology and equivariant $K$-homology. By the Baum/Connes conjecture, which holds for the Bianchi groups, we obtain the $K$-theory of their reduced $C^*$-algebras in terms of isomorphic images of the computed $K$-homology. We further find an application to Chen/Ruan orbifold cohomology. % {\it To cite this article: Alexander D. Rahm, C. R. Acad. Sci. Paris, Ser. I +++ (2011).}

math.KT↗