scispace - formally typeset
Search or ask a question
Author

D. Solitar

Other affiliations: New York University
Bio: D. Solitar is an academic researcher from York University. The author has contributed to research in topics: Free product & Free group. The author has an hindex of 16, co-authored 27 publications receiving 3199 citations. Previous affiliations of D. Solitar include New York University.

Papers
More filters
Journal ArticleDOI
TL;DR: In this article, it was shown that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the intersection of H and certain conjugates of A, b, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-NeumannNeumann group.
Abstract: We prove that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the intersection of H and certain conjugates of A, B, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-NeumannNeumann group. The particular conjugates of A, B, and U involved are given by double coset representatives in a compatible regular extended Schreier system for G modulo H. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let A and B have the property P and U have the property Q. Then it is proved that G has the property P in the following cases: P means every f.g. (finitely generated) subgroup is finitely presented, and Q means every subgroup is f.g.; P means the intersection of two f.g. subgroups is f.g., and Q means finite; P means locally indicable, and Q means cyclic. It is also proved that if A' is a f.g. normal subgroup of G not contained in U, then NU has finite index in G.

187 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.
Abstract: Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

174 citations


Cited by
More filters
Book
01 Jan 1992
TL;DR: Adeles and Ideles as discussed by the authors gave a generalization of the Strong Approximation Theorem for algebraic groups over locally compact fields and showed that the strong and weak approximations in algebraic numbers of groups are equivalent.
Abstract: (Chapter Heading): Algebraic Number Theory. Algebraic Groups. Algebraic Groups over Locally Compact Fields. Arithmetic Groups and Reduction Theory. Adeles. Galois Cohomology. Approximation in Algebraic Groups. Class Numbers andClass Groups of Algebraic Groups. Normal Structure of Groups of Rational Points of Algebraic Groups. Appendix A. Appendix B: Basic Notation. Algebraic Number Theory: Algebraic Number Fields, Valuations, and Completions. Adeles and Ideles Strong and Weak Approximation The Local-Global Principle. Cohomology. Simple Algebras over Local Fields. Simple Algebras over Algebraic Number Fields. Algebraic Groups: Structural Properties of Algebraic Groups. Classification of K-Forms Using Galois Cohomology. The Classical Groups. Some Results from Algebraic Geometry. Algebraic Groups over Locally Compact Fields: Topology and Analytic Structure. The Archimedean Case. The Non-Archimedean Case. Elements of Bruhat-Tits Theory. Results Needed from Measure Theory. Arithmetic Groups and Reduction Theory: Arithmetic Groups. Overview of Reduction Theory: Reduction in GLn(R).Reduction in Arbitrary Groups. Group-Theoretic Properties of Arithmetic Groups. Compactness of GR/GZ. The Finiteness of the Volume of GR/GZ. Concluding Remarks on Reduction Theory. Finite Arithmetic Groups. Adeles: Basic Definitions. Reduction Theory for GA Relative to GK. Criteria for the Compactness and the Finiteness of Volume of GA/GK. Reduction Theory for S-Arithmetic Subgroups. Galois Cohomology: Statement of the Main Results. Cohomology of Algebraic Groups over Finite Fields. Galois Cohomology of Algebraic Tori. Finiteness Theorems for Galios Cohomology. Cohomology of Semisimple Algebraic Groups over Local Fields and Number Fields. Galois Cohomology and Quadratic, Hermitian, and Other Forms. Proof of Theorems 6.4 and 6.6: Classical Groups. Proof of Theorems 6.4 and 6.6: Exceptional Groups. Approximation in Algebraic Groups: Strong and Weak Approximation in Algebraic Varieties. The Kneser-Tits Conjecture. Weak Approximation in Algebraic Groups. The Strong Approximation Theorem. Generalization of the Strong Approximation Theorem. Class Numbers and Class Groups of Algebraic Groups: Class Numbers of Algebraic Groups and Number of Classes in a Genus. Class Numbers and Class Groups of Semisimple Groups of Noncompact Type The Realization Theorem. Class Numbers of Algebraic Groups of Compact Type. Estimating the Class Number for Reductive Groups. The Genus Problem. Normal Subgroup Structure of Groups of Rational Points of Algebraic Groups: Main Conjecture and Results. Groups of Type An. The Classical Groups. Groups Split over a Quadratic Extension. The Congruence Subgroup Problem (A Survey). Appendices: Basic Notation. Bibliography. Index.

1,268 citations

Book ChapterDOI
01 Jan 1989

1,062 citations

Journal ArticleDOI
TL;DR: Magnusson expansion as discussed by the authors provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory (TEPT).

1,013 citations

01 Jan 2002
TL;DR: These notes were prepared by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996 and have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne.
Abstract: These notes were prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller.

913 citations

Journal ArticleDOI
TL;DR: In this article, a negative solution to the problem of Milnor concerning the degrees of growth of groups was given, which also answers a question of Day concerning amenable groups and a number of other results are obtained on residually finite finitely generated infinite 2-groups.
Abstract: This paper gives a negative solution to the problem of Milnor concerning the degrees of growth of groups. The construction also answers a question of Day concerning amenable groups. A number of other results are obtained on residually finite finitely generated infinite 2-groups. Bibliography: 51 titles.

557 citations