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The B. B. Newman Spelling Theorem [article]

Carl-Fredrik Nyberg-Brodda
2021 arXiv   pre-print
After a description of some of the theory of one-relator groups, an important part of combinatorial group theory, the natural division line into the torsion and torsion-free case for such groups is described  ...  This culminates in a statement of and general discussion about the B. B. Newman Spelling Theorem and its importance.  ...  Newman Spelling Theorem With a sufficient overview of some of the relevant theory of one-relator groups, it is now possible to present arguably the most significant result for one-relator groups with torsion  ... 
arXiv:2004.01484v2 fatcat:a7dy7yuannhpbbnwny2rsozuoa

Page 1978 of Mathematical Reviews Vol. 52, Issue 6 [page]

1976 Mathematical Reviews  
The author’s first theorem states that H is either free on the given set of generators or a one-relator group with torsion.  ...  The Baumslag-Solitar group <a, b; b-1ab=a*) is cited to show that the first and last results above cannot be extended to torsion- free one-relator groups.  ... 

Page 3535 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
Given a fixed rank-| torsion-free abelian group X, a torsion-free abelian group G is said to be X-locally free if the outer type of G is < type X and, for any prime p, G is p-divisible exactly when X is  ...  Summary: “Let M be a monoid presented by (: u = v), where u and v are words on the finite alphabet £.  ... 

Reflections on the residual finiteness of one-relator groups

Gilbert Baumslag, Charles Miller III, Douglas Troeger
2007 Groups, Geometry, and Dynamics  
work on the conjugacy and isomorphism problems for one-relator groups as a whole.  ...  For all w which do not commute with r in the ambient free group on the generators a, b, ..., the groups G.r; w/ D ha; b; : : : j r r w D r 2 i are not residually finite and have the same finite images  ...  Newman [20] proved that one-relator groups with torsion have a solvable conjugacy problem as a consequence of his so-called spelling theorem.  ... 
doi:10.4171/ggd/11 fatcat:dm4kve2ao5af5nmynxaskgcoea

Research announcement: The structure of groups with a quasiconvex hierarchy

Daniel Wise
2009 Electronic Research Announcements in Mathematical Sciences  
It follows that every quasiconvex subgroup of G is a virtual retract, and is hence separable. The results are applied to certain 3-manifold and one-relator groups.  ...  Let G be a word-hyperbolic group with a quasiconvex hierarchy. We show that G has a finite index subgroup G that embeds as a quasiconvex subgroup of a right-angled Artin group.  ...  In this case, all torsion is conjugate into W ∼ = Z n and the group is virtually torsion-free. We refer to [32] for more information on one-relator groups.  ... 
doi:10.3934/era.2009.16.44 fatcat:zhftsdi2wjf7vhwnilfasj6l5a

The universal torsion-free image of a group

Sergei D. Brodsky, James Howie
1997 Israel Journal of Mathematics  
torsion free groups.  ...  Repeating this argument, we can reduce P to a presentation for G with all relators of length exactly 3. Now apply Theorem ??. These results are best possible, as the following examples show.  ...  M is itself locally indicable, being a torsion-free one-relator group ? . Suppose K is a normal subgroup of M such that H = M =K is torsion-free.  ... 
doi:10.1007/bf02937335 fatcat:2ctopml52jbbdk3mlyxuzqv6ce

Page 476 of Mathematical Reviews Vol. 52, Issue 2 [page]

1976 Mathematical Reviews  
On the other hand, if G is a torsion-free abelian group with |G| =X, and G is not free abelian of rank &, then Ge Z. Theorem | states that if G is a solvable group of finite special rank, then Ge X.  ...  It is announced in a postscript that the author has subsequently proved that every two-generator one-relator group with torsion (TOT group), which is not a free product, has only one NEC (cf. <x, y|x?  ... 

The outer automorphism groups of two-generator, one-relator groups with torsion

Alan D. Logan
2016 Proceedings of the American Mathematical Society  
The main result of this paper is a complete classification of the outer automorphism groups of two-generator, one-relator groups with torsion.  ...  To this classification we apply recent algorithmic results of Dahmani--Guirardel, which yields an algorithm to compute the isomorphism class of the outer automorphism group of a given two-generator, one-relator  ...  Secondly, every two-generator subgroup of a one-relator group with torsion is either a free group or a two-generator, one-relator group with torsion [Pri77b] . Main theorem.  ... 
doi:10.1090/proc/13063 fatcat:eswvqqq6n5bgviyfp2up7iej3u

Page 1447 of Mathematical Reviews Vol. , Issue 82d [page]

1982 Mathematical Reviews  
By reduction to the case of ordinary one-relator groups, he proves the analogues of the freiheitssatz, Newman's spelling theorem, the torsion and conjugacy theorems, etc.  ...  The proofs depend on sharpening results on words representing torsion elements and on words representing the identity in one-relator groups. A. H. M.  ... 

Some small aspherical spaces

Eldon Dyer, A. T. Vasquez
1973 Journal of the Australian Mathematical Society  
Thus, X /Z(a) is homeomorphic to C(P'). Since X is aspherical and the projection X -> X /Z(a) = C(P') is a covering map, C(P') is aspherical.  ...  One relator presentations Let P: {x u ---,x n ; R} be a one relator presentation for the group G.  ...  A similar argument was made by Cockcoft in [2] . COROLLARY 2.3: The group G with a one relator presentation P: {x ls ---,x n ;K} is torsion free if R is not a proper power.  ... 
doi:10.1017/s1446788700015147 fatcat:mmzc5e54qbgdxj4yzop2xg42ma

Bounded rank subgroups of Coxeter groups, Artin groups and one-relator groups with torsion [article]

Ilya Kapovich, Paul Schupp
2002 arXiv   pre-print
We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.  ...  Free subgroups of one-relator groups with torsion We recall the following strengthened version of Newman's "Spelling Theorem" [45, 40] . Theorem 6.1.  ...  The class of one-relator groups with torsion is often the most tractable class of one-relator groups because of the power of Newman's Spelling Theorem [45] .  ... 
arXiv:math/0201106v1 fatcat:mgtbnmtqvjekrmpc5nsvx44xi4

Page 7335 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
{Reviewer’s remarks: (1) The paper contains numerous spelling mistakes. (2) On page 5117, reference [JOd] should read “D. L. Johnson and R. W. Odoni, Some results on symmetrically presented groups.  ...  Summary: “We prove the following result: If G is a countable group with a presentation 7, then for a positive integer m > 3 the presentation F is Tietze equivalent to a rootless slender presentation of  ... 

JSJ decompositions and polytopes for two-generator one-relator groups [article]

Giles Gardam, Dawid Kielak, Alan D. Logan
2021 arXiv   pre-print
We provide a direct connection between the 𝒵_max (or essential) JSJ decomposition and the Friedl–Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank 2.  ...  We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the 𝒵_max-JSJ decomposition of such groups.  ...  We are very grateful to Nicholas Touikan for several helpful comments and suggestions. This work has received  ... 
arXiv:2101.02193v1 fatcat:5ueheytrqbg2rnfohliya4yfsa

The isomorphism problem for plain groups is in Σ_3^𝖯 [article]

Heiko Dietrich and Murray Elder and Adam Piggott and Youming Qiao and Armin Weiß
2022 arXiv   pre-print
Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in 𝖯𝖲𝖯𝖠𝖢𝖤 when the input is given in terms of so-called virtually free presentations  ...  Here we consider the isomorphism problem for the class of plain groups, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic  ...  Acknowledgements We wish to thank the reviewers for their helpful comments and corrections.  ... 
arXiv:2110.00900v2 fatcat:5xn27nxc3bgkzp5a3u2wsucgoq

The word problem and related results for graph product groups

K. J. Horadam
1981 Proceedings of the American Mathematical Society  
The torsion and conjugacy theorems are proved for any group presented as a graph product. The graphs over which some graph product has a solvable word problem are characterised.  ...  These results generalise the known ones for HNN groups and free products with amalgamation.  ...  generated free mipotent groups), one-relator groups (see [7] ) and sixth-groups (see [4] ).  ... 
doi:10.1090/s0002-9939-1981-0609643-1 fatcat:j3rn4plogve5jfq5kwewzbe2ha
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