scholarly journals Determination of the Homology and the Cohomology of a Few Groups of Isometries of the Hyperbolic Plane

2017 ◽  
Vol 36 ◽  
pp. 65-77
Author(s):  
Nasima Akhter ◽  
Subrata Majumdar
Keyword(s):  
Hyperbolic Plane ◽  
Free Resolution ◽  
Cohomology Groups ◽  
Graph Of Groups ◽  
Finitely Presented Groups ◽  
Full Resolution ◽  
Discontinuous Groups ◽  
Groups Of Isometries ◽  
Finitely Presented

In this paper we determine the homology and the cohomology groups of two properly discontinuous groups of isometries of the hyperbolic plane having non-compact orbit spaces and the fundamental group of a graph of groups with a finite vertex groups and no trivial edges by extending Lyndon’s partial free resolution for finitely presented groups. For the first two groups, we obtain partial extensions and the corresponding homology. We also compute the corresponding cohomology groups for one of these groups. Finally we obtain homology and cohomology in all dimensions for the last of the above mentioned groups by constructing a full resolution for this group.GANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 65-77

scholarly journals On a simple method of determing the homology and the cohomology of finitely presented groups

1970 ◽  
Vol 34 (2) ◽  
pp. 103-114
Author(s):  
Subrata Majumdar ◽  
Nasima Akhter
Keyword(s):  
Linear Equations ◽  
Free Resolution ◽  
Nilpotent Groups ◽  
General Situation ◽  
Group Rings ◽  
Integral Group ◽  
Group Presentation ◽  
Simple Method ◽  
Finitely Presented Groups ◽  
Finitely Presented

In this paper the authors obtained a method of constructing free resolutions of Z for finitely presented groups directly from their presentations by extending Lyndon’s 3-term partial resolution to a full-length resolution. Authors resolutions and the method of their construction are such that free generators of the modules and the boundary homomorphisms are directly and explicitly obtained by solving of linear equations over the corresponding integral group rings, and hence these are immediately applicable for computing homology and cohomology of the groups for arbitrary coefficient modules. Authors have also described a general situation where their method is valid. The method has been used for a number of classes of group including Fuchsian groups, a few Euclidean crystallographic groups, NEC groups, the fundamental groups of a few interesting manifolds, groups of isometries of the hyperbolic plane and a few nilpotent groups of class 2. Key words: Group presentation; Free resolution; Homology; Cohomology DOI: 10.3329/jbas.v34i2.6854Journal of Bangladesh Academy of Sciences, Vol. 34, No. 2, 103-114, 2010

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 (1964), pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system (an application of methods of Britton). Proceedings of the National Academy of Sciences, vol. 53 (1965), pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 (1966), pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Annals of mathematics, ser. 2 vol. 84 (1966), pp. 49–84.

1968 ◽  
Vol 33 (2) ◽  
pp. 296-297
Author(s):  
J. C. Shepherdson
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Nauk Sssr ◽  
London Mathematical Society ◽  
National Academy Of Sciences ◽  
Doklady Akademii Nauk Sssr ◽  
Finitely Presented Groups ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

VERIFYING THAT A GROUP IS VIRTUALLY FREE

1991 ◽  
Vol 01 (03) ◽  
pp. 339-351
Author(s):  
ROBERT H. GILMAN
Keyword(s):  
Finite Index ◽  
Free Subgroup ◽  
Universal Group ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finitely Presented

This paper is concerned with computation in finitely presented groups. We discuss a procedure for showing that a finite presentation presents a group with a free subgroup of finite index, and we give methods for solving various problems in such groups. Our procedure works by constructing a particular kind of partial groupoid whose universal group is isomorphic to the group presented. When the procedure succeeds, the partial groupoid can be used as an aid to computation in the group.

scholarly journals Finitely presented groups and the Whitehead nightmare

2017 ◽  
Vol 11 (1) ◽  
pp. 291-310
Author(s):  
Daniele Ettore Otera ◽  
Valentin Poénaru
Keyword(s):  
Finitely Presented Groups ◽  
Finitely Presented
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