34 citations · 89 across the 16 of their papers we have counts for
25 papers
The 2-torsion in the Farrell--Tate cohomology of PSL(4,Z), and torsion subcomplex reduction via discrete Morse theory
Alexander D. Rahm, Anh Tuan Bui, Matthias Wendt
In the present paper, we use discrete Morse theory to provide a new implementation of torsion subcomplex reduction for arithmetic groups. This leads both to a simpler algorithm as…
On the Calegari-Venkatesh conjecture connecting modular forms spaces and algebraic K-Theory
Alexander D. Rahm, Torti Emiliano
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…
Computations regarding the torsion homology of Oeljeklaus-Toma manifolds
Dung Phuong Phan, Tuan Anh Bui, Alexander D. Rahm
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 an…
An introduction to torsion subcomplex reduction
Alexander Rahm
This survey paper introduces to a technique called Torsion Subcomplex Reduction (TSR) for computing torsion in the cohomology of discrete groups acting on suitable cell complexes.…
Bianchi's additional symmetries
Alexander Rahm
In a 2012 note in Comptes Rendus Math{é}matique, the author did try to answer a question of Jean-Pierre Serre; it has recently been announced that the scope of that answer needs an…
On Farrell-Tate cohomology of GL(3) over rings of quadratic integers
Bui Anh Tuan, Alexander D. Rahm, Matthias Wendt
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…
The distribution of rational numbers on Cantor's middle thirds set
Alexander Rahm, Noam Solomon, Tara Trauthwein +1
We give a heuristic for the number of reduced rationals on Cantor's middle thirds set, with a fixed bound on the denominator. We also describe extensive numerical computations supp…
Verification of the Quillen conjecture in the rank 2 imaginary quadratic case
Bui Anh Tuan, Alexander Rahm
We confirm a conjecture of Quillen in the case of the mod cohomology of arithmetic groups , where ${\m…
The mod 2 cohomology rings of congruence subgroups in the Bianchi groups
Ethan Berkove, Grant Lakeland, Alexander Rahm +2
We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental d…
Equivariant -homology for hyperbolic reflection groups
Jean-François Lafont, Ivonne J. Ortiz, Alexander Rahm +1
We compute the equivariant -homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological …
Genuine Bianchi modular forms of higher level, at varying weight and discriminant
Alexander Rahm, Panagiotis Tsaknias
Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Even though modern studies of Bianchi modular forms go back to the mid…
The Farrell--Tate and Bredon homology for PSL\_4(Z) via cell subdivisions
Anh Tuan Bui, Alexander Rahm, Matthias Wendt
We provide some new computations of Farrell--Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell--Tate or Bredon homology, one needs cell complexes wher…
The mod 2 cohomology rings of SL\_2 of the imaginary quadratic integers
Ethan Berkove, Alexander Rahm
We establish general dimension formulae for the second page of the equivariant spectral sequence of the action of the SL\_2 groups over imaginary quadratic integers on their associ…
A refinement of a conjecture of Quillen
Alexander Rahm, Matthias Wendt
We present some new results on the cohomology of a large scope of SL\_2-groups in degrees above the virtual cohomological dimension; yielding some partial positive results for the…
On the equivariant K-homology of PSL\_2 of the imaginary quadratic integers
Alexander Rahm
We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL\_2 of the imaginary quadratic integers), in terms of elementary numbe…
On Farrell-Tate cohomology of SL\_2 over S-integers
Alexander Rahm, Matthias Wendt
In this paper, we provide number-theoretic formulas for Farrell-Tate cohomology for SL\_2 over rings of S-integers in number fields satisfying a weak regularity assumption. These f…
Higher torsion in the Abelianization of the full Bianchi groups
Alexander D. Rahm
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 hom…
On a question of Serre
Alexander D. Rahm
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…
Accessing the cohomology of discrete groups above their virtual cohomological dimension
Alexander D. Rahm
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…
On Ruan's Cohomological Crepant Resolution Conjecture for the complexified Bianchi orbifolds
Fabio Perroni, Alexander Rahm
We give formulae for the Chen-Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups…
The homological torsion of PSL_2 of the imaginary quadratic integers
Alexander Rahm
Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reve…
Complexifiable characteristic classes
Alexander D. Rahm
We examine the topological characteristic cohomology classes of complexified vector bundles. In particular, all the classes coming from the real vector bundles underlying the compl…
Homology and K-theory of the Bianchi groups
Alexander D. Rahm
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…
On Level One Cuspidal Bianchi Modular Forms
Alexander D. Rahm, Mehmet Haluk Sengun
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 e…