Ralph Strebel - Academia.edu (original) (raw)

Papers by Ralph Strebel

Journal of Pure and Applied Algebra, Oct 1, 1988

Mathematical surveys, Nov 15, 2016

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a fi... more Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G(I; A, P) are studied which generalize the local definition of Thompson's group F in the following manner: the group G(I; A, P) consists of all orientation preserving PL-auto-homeomorphisms of the real line with support in the interval I, slopes in the multiplicative subgroup P of the positive reals and breaks in a finite subset of the additive Z[P ]-submodule A of R add. If I is the unit interval, A the subring Z[1/2] and P the cyclic group generated by the integer 2, one recovers Thompson's group F. A first aim of the monograph is to investigate in which form familiar properties of F continue to hold for these groups. Here is a sample: the group F is known to be simple by abelian; we shall see that the group G(I; A, P) has the same form if I is a compact interval with endpoints in A and that it is, in general, simple by soluble of derived length at most 3. Main aims of the monograph are the determination of isomorphisms among the groups G(I; A, P) and the study of their automorphism groups. Complete answers are obtained if the group P is not cyclic or if the interval I is the full line. In the case of automorphisms, these answers are the only known results about Aut G(I; A, P), save for findings due to M. G. Brin and to Brin-Guzmán that deal with special cases not covered by the monograph. x Preface again index 2 in Aut G and the automorphism by conjugation by the reflection t → b − t represents the coset Aut G Inn G. If, finally, I is a half line with endpoint in A, then Aut G = Inn G and the group G(I; A; Aut o (A)) is complete. The assumptions that P is all of Aut o (A) and that it is not cyclic hold, in particular, for PL o (R) = G(R; R, R × >0), a group considered in Corollary 31 of [McC78]. Corollary 9 thus puts McCleary's result into a larger context. There is a second aspect of Corollary 9 that deserves mention, namely the fact that, given parameters I, A and P as in the corollary, Aut G(I; A, P) has a subgroup Aut o G(I; A, P) of index at most 2, which is a subgroup of the group G(I; A, Aut o (A)). Here Aut o G(I; A, P) denotes the subgroup of increasing automorphisms, the automorphisms induced by orientation preserving auto-homeomorphisms. Some consequences of this fact are spelled out by Corollary E17.8. Supplement E17.3 allows one also to determine when two intervals [0, b 1 ] and [0, b 2 ] lead to isomorphic groups. The answer depends once more on the automorphism group of A, more precisely on its index 2 subgroup Aut o (A).

arXiv: Group Theory, 2018

In this article, I study some classes of finitely presented groups with the aim of finding out wh... more In this article, I study some classes of finitely presented groups with the aim of finding out whether the maximal metabelian quotients of the members of these classes admit finite presentations. The considered classes include those of soluble groups, of one-relator or knot groups, and of Artin groups.

Canadian Mathematical Bulletin, 1981

It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and ... more It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalities We prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.

In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1... more In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1981), no. 1-2, pages 25-45; MR 617288) A. Lubotzky and L. van den Dries raise the question whether a finitely generated, residually finite group is necessarily free if the rank function on its subgroups of finite index satisfies Schreier's well-known index rank relation (see Question 2 on p. 34). I answered this question in 1980 but, so far, I have not published my answer. This note fills the omission; it gives an amended and abridged version of my original proof.

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a fini... more We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a fi... more Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G are studied which depend on three parameters I, A, and P and which generalize the local definition of Thompson's group F thus: G consists of all orientation preserving PL-homeomorphisms of the real line with supports in the interval I, slopes in the multiplicative subgroup P of the positive reals and breaks in a finite subset of the additive P submodule A of R. A first aim of the monograph is to investigate in which form familiar properties of F continue to hold for these groups. Main aims of the monograph are the determination of isomorphisms among the groups G and the study of the...

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a ge... more A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Bieri and R. Strebel in 1980.

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a ge... more A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Bieri and R. Strebel in 1980.

Journal of the London Mathematical Society, 1982

Inventiones Mathematicae, 1987

Skeletal integrity in humans and animals is maintained by daily mechanical loading. It has been w... more Skeletal integrity in humans and animals is maintained by daily mechanical loading. It has been widely accepted that osteocytes function as mechanosensors. Many biochemical signaling molecules are involved in the response of osteocytes to mechanical stimulation. The aim of this study was to identify genes involved in the translation of mechanical stimuli into bone formation. The four-point bending model was used to induce a single period of mechanical loading on the right tibia, while the contra lateral left tibia served as control. Six hours after loading, the effects of mechanical loading on gene-expression were determined with microarray analysis. Protein expression of differentially regulated genes was evaluated with immunohistochemistry. Nine genes were found to exhibit a significant differential gene expression in LOAD compared to control. MEPE, Garnl1, V2R2B, and QFG-TN1 olfactory receptor were up-regulated, and creatine kinase (muscle form), fibrinogen-B beta-polypeptide, monoamine oxidase A, troponin-C and kinesin light chain-C were down-regulated. Validation with real-time RT-PCR analysis confirmed the up-regulation of MEPE and the down-regulation of creatine kinase (muscle form) and troponin-C in the loaded tibia. Immunohistochemistry showed that the increase of MEPE protein expression was already detectable six hours after mechanical loading. In conclusion, these genes probably play a role during translation of mechanical stimuli six hours after mechanical loading. The modulation of MEPE expression may indicate a connection between bone mineralization and bone formation after mechanical stimulation.

Archiv der Mathematik, 1981

Archiv der Mathematik, 1983

We describe various classes of infinitely presented groups that are condensation points in the sp... more We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

arXiv: Group Theory, 2018

In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1... more In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1981), no. 1-2, pages 25-45; MR 617288) A. Lubotzky and L. van den Dries raise the question whether a finitely generated, residually finite group is necessarily free if the rank function on its subgroups of finite index satisfies Schreier's well-known index rank relation (see Question 2 on p. 34). I answered this question in 1980 but, so far, I have not published my answer. This note fills the omission; it gives an amended and abridged version of my original proof.

Mathematical Surveys and Monographs, 2016

Journal of Pure and Applied Algebra, Oct 1, 1988

Mathematical surveys, Nov 15, 2016

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a fi... more Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G(I; A, P) are studied which generalize the local definition of Thompson's group F in the following manner: the group G(I; A, P) consists of all orientation preserving PL-auto-homeomorphisms of the real line with support in the interval I, slopes in the multiplicative subgroup P of the positive reals and breaks in a finite subset of the additive Z[P ]-submodule A of R add. If I is the unit interval, A the subring Z[1/2] and P the cyclic group generated by the integer 2, one recovers Thompson's group F. A first aim of the monograph is to investigate in which form familiar properties of F continue to hold for these groups. Here is a sample: the group F is known to be simple by abelian; we shall see that the group G(I; A, P) has the same form if I is a compact interval with endpoints in A and that it is, in general, simple by soluble of derived length at most 3. Main aims of the monograph are the determination of isomorphisms among the groups G(I; A, P) and the study of their automorphism groups. Complete answers are obtained if the group P is not cyclic or if the interval I is the full line. In the case of automorphisms, these answers are the only known results about Aut G(I; A, P), save for findings due to M. G. Brin and to Brin-Guzmán that deal with special cases not covered by the monograph. x Preface again index 2 in Aut G and the automorphism by conjugation by the reflection t → b − t represents the coset Aut G Inn G. If, finally, I is a half line with endpoint in A, then Aut G = Inn G and the group G(I; A; Aut o (A)) is complete. The assumptions that P is all of Aut o (A) and that it is not cyclic hold, in particular, for PL o (R) = G(R; R, R × >0), a group considered in Corollary 31 of [McC78]. Corollary 9 thus puts McCleary's result into a larger context. There is a second aspect of Corollary 9 that deserves mention, namely the fact that, given parameters I, A and P as in the corollary, Aut G(I; A, P) has a subgroup Aut o G(I; A, P) of index at most 2, which is a subgroup of the group G(I; A, Aut o (A)). Here Aut o G(I; A, P) denotes the subgroup of increasing automorphisms, the automorphisms induced by orientation preserving auto-homeomorphisms. Some consequences of this fact are spelled out by Corollary E17.8. Supplement E17.3 allows one also to determine when two intervals [0, b 1 ] and [0, b 2 ] lead to isomorphic groups. The answer depends once more on the automorphism group of A, more precisely on its index 2 subgroup Aut o (A).

arXiv: Group Theory, 2018

In this article, I study some classes of finitely presented groups with the aim of finding out wh... more In this article, I study some classes of finitely presented groups with the aim of finding out whether the maximal metabelian quotients of the members of these classes admit finite presentations. The considered classes include those of soluble groups, of one-relator or knot groups, and of Artin groups.

Canadian Mathematical Bulletin, 1981

It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and ... more It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalities We prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.

In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1... more In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1981), no. 1-2, pages 25-45; MR 617288) A. Lubotzky and L. van den Dries raise the question whether a finitely generated, residually finite group is necessarily free if the rank function on its subgroups of finite index satisfies Schreier's well-known index rank relation (see Question 2 on p. 34). I answered this question in 1980 but, so far, I have not published my answer. This note fills the omission; it gives an amended and abridged version of my original proof.

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a fini... more We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a fi... more Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a peculiar infinite presentation and has a local definition which implies that F is dense in the topological group of all orientation preserving homeomorphisms of the unit interval. In this monograph groups G are studied which depend on three parameters I, A, and P and which generalize the local definition of Thompson's group F thus: G consists of all orientation preserving PL-homeomorphisms of the real line with supports in the interval I, slopes in the multiplicative subgroup P of the positive reals and breaks in a finite subset of the additive P submodule A of R. A first aim of the monograph is to investigate in which form familiar properties of F continue to hold for these groups. Main aims of the monograph are the determination of isomorphisms among the groups G and the study of the...

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a ge... more A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Bieri and R. Strebel in 1980.

A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a ge... more A recipe for obtaining finitely presented abelian-by-nilpotent groups is given. It relies on a geometric procedure that generalizes the construction of finitely presented metabelian groups introduced by R. Bieri and R. Strebel in 1980.

Journal of the London Mathematical Society, 1982

Inventiones Mathematicae, 1987

Skeletal integrity in humans and animals is maintained by daily mechanical loading. It has been w... more Skeletal integrity in humans and animals is maintained by daily mechanical loading. It has been widely accepted that osteocytes function as mechanosensors. Many biochemical signaling molecules are involved in the response of osteocytes to mechanical stimulation. The aim of this study was to identify genes involved in the translation of mechanical stimuli into bone formation. The four-point bending model was used to induce a single period of mechanical loading on the right tibia, while the contra lateral left tibia served as control. Six hours after loading, the effects of mechanical loading on gene-expression were determined with microarray analysis. Protein expression of differentially regulated genes was evaluated with immunohistochemistry. Nine genes were found to exhibit a significant differential gene expression in LOAD compared to control. MEPE, Garnl1, V2R2B, and QFG-TN1 olfactory receptor were up-regulated, and creatine kinase (muscle form), fibrinogen-B beta-polypeptide, monoamine oxidase A, troponin-C and kinesin light chain-C were down-regulated. Validation with real-time RT-PCR analysis confirmed the up-regulation of MEPE and the down-regulation of creatine kinase (muscle form) and troponin-C in the loaded tibia. Immunohistochemistry showed that the increase of MEPE protein expression was already detectable six hours after mechanical loading. In conclusion, these genes probably play a role during translation of mechanical stimuli six hours after mechanical loading. The modulation of MEPE expression may indicate a connection between bone mineralization and bone formation after mechanical stimulation.

Archiv der Mathematik, 1981

Archiv der Mathematik, 1983

We describe various classes of infinitely presented groups that are condensation points in the sp... more We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

arXiv: Group Theory, 2018

In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1... more In "Subgroups of free profinite groups and large subfields of Q" (Israel J. Math. 39 (1981), no. 1-2, pages 25-45; MR 617288) A. Lubotzky and L. van den Dries raise the question whether a finitely generated, residually finite group is necessarily free if the rank function on its subgroups of finite index satisfies Schreier's well-known index rank relation (see Question 2 on p. 34). I answered this question in 1980 but, so far, I have not published my answer. This note fills the omission; it gives an amended and abridged version of my original proof.

Mathematical Surveys and Monographs, 2016