
Courses
Courses
Choosing a course is one of the most important decisions you'll ever make! View our courses and see what our students and lecturers have to say about the courses you are interested in at the links below.

University Life
University Life
Each year more than 4,000 choose NUI Galway as their University of choice. Find out what life at NUI Galway is all about here.

About NUI Galway
About NUI Galway
Since 1845, NUI Galway has been sharing the highest quality teaching and research with Ireland and the world. Find out what makes our University so special – from our distinguished history to the latest news and campus developments.

Colleges & Schools
Colleges & Schools
NUI Galway has earned international recognition as a researchled university with a commitment to top quality teaching across a rang of key areas of expertise.

Research
Research
NUI Galway’s vibrant research community take on some of the most pressing challenges of our times.

Business & Industry
Guiding Breakthrough Research at NUI Galway
We explore and facilitate commercial opportunities for the research community at NUI Galway, as well as facilitating industry partnership.

Alumni, Friends & Supporters
Alumni, Friends & Supporters
There are over 90,000 NUI Galway graduates Worldwide, connect with us and tap into the online community.

Community Engagement
Community Engagement
At NUI Galway, we believe that the best learning takes place when you apply what you learn in a real world context. That's why many of our courses include work placements or community projects.
Mathematics, Statistics, And Applied Mathematics
Dr Alexander Rahm Doctor of Mathematics
Contact Details

Biography:
2001, Abitur (degree for entering universities in Germany), MainTaunusSchule (Gymnasium) Hofheim am Taunus, Germany 2003, Vordiplom Physik, Universität Göttingen, Germany 2004, Licence de Mathématiques, Université Montpellier II, France 2007, Diplom Mathematik, Mathematisches Institut, Universität Göttingen, Germany 2010, Binational doctoral degree (Cotutelle de thèse), with the advisors Prof. Dr. Philippe ElbazVincent , at the Institut Fourier, Université Joseph Fourier Grenoble, France and Prof. Dr. Thomas Schick, Mathematisches Institut, Universität Göttingen, Germany. Supported by a research grant of the Ministère de l'Enseignement Supérieur et de la Recherche, and partially supported by DFHUFA grant CT2607I and by a DAAD (German Academic Exchange Service) grant. The intensive computations award C3I has been attributed to this thesis. The thesis was defended October 15th, 2010, Grenoble. It can be downloaded from the server TEL of the CNRS. The iterated advisors of my advisors can be found in my mathematics genealogy graph. I am grateful for a Research stay at the I3M Montpellier during my doctoral studies.Postdoctoral scientific research:20102011, Research fellow of the Weizmann Institute of Science, Department of Mathematics.Mentor: Prof. Dr. Stephen S. Gelbart. 20112013, Research fellow of the Irish Research Council for Science, Engineering and Technology. Affiliated to the National University of Ireland at Galway, Department of Mathematics. Mentor: Prof. Dr. Graham J. Ellis. Since 2013, Lecturer of Mathematics at National University of Ireland, Galway. 
Research Interests
Algebra, Geometry and Topology, for instance:  Classifying Space for proper actions;  Discrete subgroups of Lie groups;  Discontinuous groups of transformations;  Isometries of Hyperbolic Space;  And many others. A detailed description of my current research projects is given here. You can watch a video of a talk I gave at a Banff workshop on the Torsion Subcomplex Reduction technique that I have introduced. Torsion Subcomplex Reduction Torsion Subcomplex Reduction is a technique which I have developed for the computation of the torsion in the cohomology of discrete groups. Initially, I did develop this technique specifically for the Bianchi groups; later I did put it into a more general framework. Some elements of this technique have beforehand already been used as ad hoc tricks by Soulé, and later by Mislin and Henn. The advantage of using a systematic technique rather than a set of adhoc tricks, is that instead of merely helping isolated example calculations, it becomes possible to find general formulae, as I did for instance for all of the Bianchi groups for all of the cohomology above the virtual cohomological dimension. On the Bianchi groups, I have successfully transplanted my technique to the Bredon homology computations for topological Ktheory, where so far only isolated examples had been calculated, and I expect to find general formulae for large classes of groups relevant for the BaumConnes conjecture. This promises a big impact both for cases in which the BaumConnes conjecture has been proved, and where hence the formulae will extend to Operator Ktheory, and for cases in which the BaumConnes conjecture is still open, and where progress on the Operator Ktheory side could be made. It is convenient to give some examples of where the technique of Torsion Subcomplex Reduction has already produced good results: • The Bianchi groups, • The Coxeter groups, • The SL2 groups over arbitrary number rings. The Bianchi groups. In the case of the Bianchi groups (the PSL2 groups over rings of imaginary quadratic integers), the Torsion Subcomplex Reduction technique has permitted me to find a description of the cohomology ring of these groups in terms of elementary numbertheoretic quantities; see my paper Accessing the cohomology of discrete groups above their virtual cohomological dimension, J. Algebra 404 (2014), 152175. MR3177890. The decisive step has been to extract, using Torsion Subcomplex Reduction, the essential information about the geometric models, and then to detach this information completely from the model. I was hence able to show that this information is contained in conjugacy classes graphs, which I construct for this purpose for an arbitrary group from its system of conjugacy classes of finite subgroups. The Coxeter groups. Recall that the Coxeter groups are generated by reflections; and their homology consists uniquely of torsion. Torsion Subcomplex Reduction hence allows to obtain all of the homology of all of the tetrahedral Coxeter groups at all odd prime numbers, both in a general formula and in terms of explicit tables [see my above quoted paper]. The SL_2 groups over arbitrary number rings. In collaboration with Matthias Wendt, I have established formulae for the FarrellTate cohomology with odd torsion coefficients for all groups SL2 (A), where A is a ring of Sintegers in an arbitrary number field; see my joint paper with Matthias Wendt, On FarrellTate cohomology of SL_2 over Sintegers, https://hal.archivesouvertes.fr/hal01081081, Wendt has furthermore extended this to the cases where A is the ring of functions on a smooth affine curve over an algebraically closed field. These two results together have allowed Wendt to find a refined version of the Quillen conjecture, which keeps track of all the types of known counterexamples to the original Quillen conjecture; see our joint paper A refinement of a conjecture of Quillen, accepted for publication in Comptes Rendus Mathématique of the Académie des Sciences  Paris. So if there does not exist any counterexample of completely new type to the original Quillen conjecture, then the QuillenWendt conjecture must be true. 
Research Projects
Project: 30th Summer Conference on Topology and its Applications ( 30thSCTA) Start/End Dates: 23JUN15 / 26JUN15 
Project: Torsion Subcomplex Reduction Research at Institut des Hautes Études Scientifiques (IHES) ( IHESvisit) Start/End Dates: / 
Project: Constructing explicit elements in algebraic Ktheory ( CEEalgKtheory) Start/End Dates: 17APR16 / 30APR16 
Project: Conference workshop Groups in Galway 2015 ( GroupsInGalway2015) Start/End Dates: 22MAY15 / 23MAY15 
Project: Conference workshop "Groups in Galway 2014" ( Groups in Galway) Start/End Dates: 23MAY14 / 24MAY14 
Project: Dimension computation of Bianchi modular forms ( ngmat003b) Start/End Dates: 01JUL12 / 30JUN14 
Peer Reviewed Journals
Alexander D. Rahm and Matthias Wendt (2015) 'A refinement of a conjecture of Quillen'. Comptes Rendus Mathematique, . [Details] 
Alexander D. Rahm (2014) 'Accessing the cohomology of discrete groups above their virtual cohomological dimension'. Journal Of Algebra, Accepted for publication . [Details] 
Alexander D. Rahm (2014) 'Complexifiable characteristic classes'. Journal of Homotopy and Related Structures, . [Details] 
Rahm, Alexander D. (2014) 'The subgroup measuring the defect of the abelianization of $\rm SL_2(\Bbb Z[i])$'. J. Homotopy Relat. Struct, 9 (2):257262. [DOI] [Details] 
Rahm, AD (2013) 'The homological torsion of PSL_2 of the imaginary quadratic integers'. Transactions Of The American Mathematical Society, 365 :16031635. [DOI] [Details] 
Rahm, AD (2013) 'Higher torsion in the Abelianization of the full Bianchi groups'. Lms Journal Of Computation And Mathematics, 16 :344365. [DOI] [Details] 
A.~D.~Rahm (2013) 'The subgroup measuring the defect of the Abelianization of SL_2(Z[i])'. Journal of Homotopy and Related Structures, . [ARAN Link] [Details] 
A.~D.~Rahm and M.~H.~Sengun (2013) 'On Level One Cuspidal Bianchi Modular Forms'. Lms Journal Of Computation And Mathematics, 16 :187199. [DOI] [ARAN Link] [Details] 
Rahm, AD (2012) 'On a question of Serre'. Comptes Rendus Mathematique, 350 :741744. [DOI] [ARAN Link] [Details] 
Rahm, Alexander D. (2011) 'Homology and $K$theory of the Bianchi groups'. C. R. Math. Acad. Sci. Paris, 349 (1112):615619. [DOI] [ARAN Link] [Details] 
Rahm, Alexander D. and Fuchs, Mathias (2011) 'The integral homology of $\rm PSL_2$ of imaginary quadratic integers with nontrivial class group'. J. Pure Appl. Algebra, 215 (6):14431472. [DOI] [ARAN Link] [Details] 
Conference Paper
Alexander D. Rahm (2007) Characteristic classes of complexified bundles. Lille 2007: Conference Paper [Details] 
Research Paper
Rob de Jeu and Alexander D. Rahm (2014) The image of the BorelSerre bordification in algebraic Ktheory. Research Paper [Details] 
Alexander D. Rahm and Matthias Wendt (2014) On FarrellTate cohomology of SL_2 over Sintegers. Research Paper [Details] 
Ethan Berkove and Alexander D. Rahm (2014) The mod 2 cohomology rings of SL$_2$ of the imaginary quadratic integers. Research Paper [Details] 
Alexander D. Rahm (2013) Chen/Ruan orbifold cohomology of the Bianchi groups. Research Paper [Details] 
Research Seminar
Alexander D. Rahm (2015) Pure Mathematics Colloquium of University of Southampton: Bredon homology calculation techniques. Research Seminar [Details] 
Alexander D. Rahm (2015) Séminaire Arithmétique et Théorie de l'Information (ATI) of the Institut de Mathématiques de Luminy: Méthodes cohomologiques pour les formes modulaires de Bianchi. Research Seminar [Details] 
Alexander D. Rahm (2015) Algebra Seminar, University of Glasgow: From a question of Serre to the QuillenWendt conjecture. Research Seminar [Details] 
Alexander D. Rahm (2015) Seminar talk at Universitaet Giessen, Germany. Research Seminar [Details] 
Alexander D. Rahm (2014) Seminar talk at Université de Neuchâtel: An access to Bredon homology. Research Seminar [Details] 
Alexander D. Rahm (2014) Seminar on Euler Systems/Arithmetic Geometry, U. DuisburgEssen: The image of the BorelSerre bordification in algebraic Ktheory. Research Seminar [Details] 
Alexander D. Rahm (2013) Séminaire Algèbre et topologie, IRMA Strasbourg: Calculs efficaces au dessus de la dimension cohomologique virtuelle. Research Seminar [Details] 
Alexander D. Rahm (2013) Seminar of the Institute of Physics and Mathematics at Universidad Austral de Chile (UACh), Valdivia: Group Actions and Orbit Spaces. Research Seminar [Details] 
Alexander D. Rahm (2013) Seminar talk at Universitaet Giessen, Germany: Determination of torsion invariants of arithmetic groups. Research Seminar [Details] 
Alexander D. Rahm (2012) Seminar talk at Vrije Universiteit Amsterdam: The arithmetic of symmetric spaces. Research Seminar [Details] 
Alexander D. Rahm (2012) Seminar talk at Universität Freiburg: Reduktion von Torsionsunterkomplexen und Konjugiertenklassengraphen. Research Seminar [Details] 
Alexander D. Rahm (2012) Seminar talk at University College Dublin: Torsion subcomplex reduction and conjugacy classes graphs. Research Seminar [Details] 
Alexander D. Rahm (2012) Séminaire du laboratoire LSIIT Strasbourg: Homologie calculatoire de groupes modulaires. Research Seminar [Details] 
Alexander D. Rahm (2011) Seminar on Representation Theory and Algebraic Geometry, W.I.S.: Numbertheoretic formulae for the homological torsion of the Bianchi groups. Research Seminar [Details] 
Alexander D. Rahm (2011) Seminar talk at Technische Universität Darmstadt: Polyhedral models for arithmetic groups. Research Seminar [Details] 
Alexander D. Rahm (2011) Séminaire Théorie des Nombres de Bordeaux: La conjecture de Baum/Connes  un accès explicite. Research Seminar [Details] 
Alexander D. Rahm (2011) Séminaire Algèbre et topologie, IRMA Strasbourg: Modèles cellulaires pour des groupes arithmétiques. Research Seminar [Details] 
Alexander D. Rahm (2011) Hebrew University of Jerusalem, Amitsur Algebra seminar: Stringy orbifolds. Research Seminar [Details] 
Alexander D. Rahm (2011) GdT. Géométrie hyperbolique et surfaces de Riemann, Univ. Paris 7, Inst. Math. Jussieu: La torsion homologique de SL_2 d'anneaux d'entiers. Research Seminar [Details] 
Alexander D. Rahm (2010) Seminar on Representation Theory and Algebraic Geometry, W.I.S.: (Co)homologies and Ktheory of Bianchi groups using computational geometric models. Research Seminar [Details] 
Alexander D. Rahm (2009) Séminaire Algèbre et Géométries, Grenoble: Domaines fondamentaux pour groupes de matrices sur des entiers quadratiques imaginaires. Research Seminar [Details] 
Alexander D. Rahm (2009) Seminar talk at National University of Ireland at Galway: The integral homology of PSL_2 of imaginary quadratic integers with nontrivial class group. Research Seminar [Details] 
Alexander D. Rahm (2009) Research Seminar Comp. Algebra & Number Theory, Düsseldorf: The integral homology of PSL_2 of imaginary quadratic integers. Research Seminar [Details] 
Alexander D. Rahm (2009) Conférence de l'Institut Fourier, Autrans: Homologie de groupes discrets d'isométries de l'espace hyperbolique. Research Seminar [Details] 
Alexander D. Rahm (2009) Séminaire Théorie des Nombres de Montpellier. Research Seminar [Details] 
Alexander D. Rahm (2009) Alg., Géom. Alg., Top. Alg., I3M Montpellier: Homologie à coefficients entiers de PSL_2 sur des anneaux d'entiers quadratiques imaginaires. Research Seminar [Details] 
Alexander D. Rahm (2006) Seminar talk at Math. Institut, Universität Tübingen: The Lefschetz hyperplane intersection theorem. Research Seminar [Details] 
Alexander D. Rahm (2006) Séminaire Gaston Darboux, I3M Montpellier: Variétés pseudoriemanniennes et réductions de fibrés principaux. Research Seminar [Details] 
Thesis
Alexander D. Rahm (2010) (Co)homologies et Kthéorie de groupes de Bianchi par des modèles géométriques calculatoires. Thesis [ARAN Link] [Details] 
Honours and Awards
Year: 2010. Title: Certificat de Compétences en Calcul Intensif (C3I) 
Associations
Association: Irish Mathematical Society (IMS), Function/Role: Member 
Association: European Mathematical Society (EMS), Function/Role: Member 
Association: Société Mathématique de France (SMF), Function/Role: Member 
Association: Deutsche MathematikerVereinigung (DMV), Function/Role: Member 
Conference Contributions
Alexander D. Rahm (2014) Lecture Bianchi modular forms of varying discriminant, level and weight. [Invited Lecture], Workshop on Bianchi and Siegel modular forms, University of Sheffield , 14JUL14  16JUL14. 
Alexander D. Rahm (2015) A software for computations on the Dynamics and Topology of the Bianchi groups. [Invited Oral Presentation], DyToComp 2015  Dynamics, Topology and Computations, Mathematical Research and Conference Center in Będlewo, Poland , 15JUN15  20JUN15. 
Alexander D. Rahm (2015) Bianchi groups computation with PARI. [Invited Oral Presentation], Atelier PARI/GP 2015, Université de Bordeaux , 12JAN15  16JAN15. 
Alexander D. Rahm (2012) Closing Talk: A new technique to extract invariants of matrix groups. [Keynote Address], 6th de Brun Workshop Linear Algebra and Matrix Theory: connections, applications and computations, NUI Galway , 03DEC12  07DEC12. 
Alexander D. Rahm (2012) Accessing the FarrellTate cohomology of discrete groups. [Invited Oral Presentation], Torsion in the homology of arithmetic groups: geometry, arithmetic, and computation, Banff International Research Station for Mathematical Innovation and Discovery, Canada , 01JUL12  06JUL12. 
Alexander D. Rahm (2011) Implications of hyperbolic geometry to operator Ktheory of artihmetic groups. [Invited Oral Presentation], London Mathematical Society  EPSRC Durham Symposium: Geometry and Arithmetic of Lattices, Durham, England , 04JUL11  14JUL11. 
Alexander D. Rahm (2009) The integral homology of PSL_2 of imag. quad. int. of class number two. [Invited Oral Presentation], 17th öMG Congress and Annual DMV Meeting, Graz, Austria, TU Graz, Autria , 20SEP09  25JAN09. 
Alexander D. Rahm (2007) Characteristic classes of complexified bundles. [Invited Oral Presentation], Summer School in Algebraic Topology: Sheaf theoretic methods in the theory of characteristic classes, Université Lille 1 , 11JUN07  15JUN07. 
Alexander D. Rahm (2006) Characteristic classes of real generator bundles. [Invited Oral Presentation], Young Topologists and new topology at the Banach Mathematical Center and Conference Center, Bedlewo, Poland , 28AUG06  02SEP06. 
Committees
Committee : Research Matters, NUI Galway, Editorial Board 
Committee : Research Committee of the Senate of Universität Göttingen 
Committee : Strategy Committee of the Senate of Universität Göttingen 
Committee : Research Committee of the School of Mathematics, Statistics and Applied Mathematics 
Committee : Outreach & Communication Committee of the School of Mathematics, Statistics and Applied Mathematics 
Employment
Employer: Irish Research Council for Science, Engineering and Technology Position: PosrDoctoral Research Fellow (EMPOWER scheme) 
Employer: Weizmann Institute of Science  Department of Mathematics (WISDoM) Position: Postdoctoral Research Fellow 
Employer: National University of Ireland, Galway Position: Lecturer of Mathematics 
Languages
English: 
French: 
Italian: 
German: 
Teaching Interests
Lecturing activities: Summer semester 2014/2015: Cryptography. Winter semester 2014/2015: Quantitative Techniques for Business and Mathematical and Logical Aspects of Computing. Summer semester 2013/2014: Calculus II and Cryptography. Winter semester 2013/2014: Quantitative Techniques for Business and Mathematical and Logical Aspects of Computing. Summer semester 2012/2013: Lecture series "Discrete groups and torsion subcomplexes" at the Graduiertenkolleg Cohomological Methods in Geometry, University of Freiburg, Germany. Winter semester 2012/2013: Quantitative Techniques for Business. 2011/2012: Graduate course in Arithmetic. Further teaching, administration and social activities of Alexander D. Rahm
Homepage of Alexander D. Rahm 
Recent Postgraduates
Currently, supervising the PhD thesis of Daher Waly Freh AlBaydli: Algebraic topology and characteristic classes of the Bianchi groups (jointly supervised with Dr. Emil Sköldberg)
Currently, supervising the Master Thesis of Katherine Wilkie:
Academic year 2013/2014, supervised the Final Year thesis of Sarah Morahan:
Academic year 2013/2014, examined the Final Year thesis of Damien Hurney:
Diplom thesis of Hendrik Demmer (defended 2011, Universität Göttingen), In his Diplom thesis, Hendrik Demmer studied the modular group: the quotient of the group SL_{2} (Z) by its centre. He constructed a cellular model for the action of the modular group, as a bridge between the classical geometrical model  the modular tree of Serre  and Kulkarni's arithmetic model. The latter admits the advantage that for every subgroup of finite index in the modular group, a Farey symbol can be computed in an efficient way, containing the essential information about the group structure. Demmer's model is a graph, constituting the 1skeleton of a twodimensional cell complex dual to the modular tree. Its set of edges is the set of elements of the modular group itself; and its 0skeleton is the projective line over the rational numbers, to which Hendrik Demmer lends additional arithmetic structure as the set of "SternBrocot fractions". This enables Demmer to compute the Farey symbol of a subgroup from a fundamental domain for it in the modular tree. To arrive there, he shows that he can choose fundamental domains in the modular tree particularly appropriately. Demmer's fundamental domains incorporate the bifurcation points into their interior and hence avoid that a bifurcation in the quotient graph becomes visible only after carrying out the identifications. Also, bifurcations of the modular tree are either incorporated entirely into Demmer's fundamental domain, or as a single edge in the case that the orbit of the latter contains the whole bifurcation. By the strictness concerning the edges, it is achieved that the number of edges of such a fundamental domain equals the index of the concerned subgroup. The information attached to the corners in terms of SternBrocot fractions now allows to compute the identifications among the endpoints of the fundamental domain as matrices, via an algorithm conceived by Hendrik Demmer. Furthermore, Hendrik Demmer has elaborated algorithms for the conversion between matrices of the modular group and words in the free product (isomorphic to the latter group) of the two finite groups of orders 2 and 3. Advice for postdoctoral researchers:
Since 2012, providing support for the project 
Internal Collaborators
With Graham Ellis and Tuan Anh Bui (both National University of Ireland at Galway) , I incorporate cell complexes into HAP, the Homological Algebra Programming package of GAP. An example of the results obtained with Ellis is included in my article The homological torsion of PSL_2 of the imaginary quadratic integers, Transactions of the AMS, volume 365 (2013), pp. 16031635. Based on these calculations, I have used the methods of my joint paper The mod 2 cohomology rings of SL_2 of the imaginary quadratic integers, preprint joint with Ethan Berkove, to successfully obtain a cohomology ring which plays an important role in a project of Gael Collinet (Universite de Strasbourg), which is being pursued by Tuan Anh Bui. 
External Collaborators
With Ethan Berkove (Lafayette College, Pennsylvania), I have written a preprint, The mod 2 cohomology rings of SL2 of the imaginary quadratic integers, http://hal.archivesouvertes.fr/hal00769261 conditionally accepted for publication. A joint paper extending these investigations to congruence subgroups is in advanced progress together with Ethan Berkove and Grant Lakeland (East Illinois University). __________________________________________________ With Matthias Wendt (Universitaet DuisburgEssen), I apply the techniques of my paper Accessing the cohomology of discrete groups above their virtual cohomological dimension, Journal of Algebra, Volume 404, 15 February 2014, Pages 152175, to large classes of arithmetic SL_n groups. Two joint papers have been written up so far: A refinement of a conjecture of Quillen, joint with Matthias Wendt, accepted for publication at Comptes Rendus Mathématique de l’Académie des Sciences, https://hal.archivesouvertes.fr/hal01083049, and On FarrellTate cohomology of SL_2 over Sintegers, joint with Matthias Wendt, recently submitted, https://hal.archivesouvertes.fr/hal01081081 ____________________________________ With Rob de Jeu (VU University Amsterdam), I search for elements in H_3(GL_2 (Q(sqrt{−m}))) related to the algebraic Ktheory of imaginary quadratic rings. Rob de Jeu has found a method to relate these elements to geometric images coming from the Bianchi groups. I have determined these images using my answer to a question of Serre, which had been open for 40 years. See our recent joint preprint, http://hal.archivesouvertes.fr/hal00975454 For an extension of this research project, I have been granted the MFOhosted research project Constructing explicit elements in algebraic Ktheory, joint with Rob de Jeu, Herbert Gangl (University of Durham) and Dan Yasaki (University of North Carolina at Greensboro) at Mathematisches Forschungsinstitut Oberwolfach (MFO), April 17 to 30, 2016. ___________________________________________ With Mehmet Haluk Sengün (University of Sheffield), I search for modular forms which cannot be obtained as lifts with the Langlands base change procedure, see our paper On Level One Cuspidal Bianchi Modular Forms, joint with Mehmet Haluk Sengün LMS Journal of Computation and Mathematics, volume 16 (2013), pp. 187199. Currently, we are evaluating the results from numerical experiments for which I have obtained an ICHEC grant of 900,000 processor hours (electricity and material replacement costs of 30,000 Euro). We are preparing these results for publication in the LMFDB database. ______________________________________________ With Fabio Perroni (University of Trieste), I study Ruan’s conjecture on crepant resolutions for stringy orbifold cohomology. I have recently submitted a paper about this, Chen/Ruan orbifold cohomology of the Bianchi orbifolds, http://hal.archivesouvertes.fr/hal00627034/ _________________________________________________ With Nicolas Bergeron (Université Paris 6) and Aurel Page (University of Warwick) (link to his previous homepage), I study the asymptotics of the Abelianization of the Bianchi groups, based on my paper Higher torsion in the Abelianization of the full Bianchi groups, LMS J. of Computation and Mathematics. __________________________________________________ With Ruben SanchezGarcia (University of Southampton), JeanFrancois Lafont (Ohio State University) and Ivonne Ortiz (Miami University, Oxford, Ohio), I compute the Bredon homology and equivariant Khomology of all Coxeter tetrahedral groups, as well as more Coxeter hyperbolic groups, using an adaptation to Bredon homology of my Torsion Subcomplex Reduction technique. 