W. D. Neumann and M. Shapiro :
“Equivalent automatic structures and their boundaries Int. J. Algebra Comput.
2 : 4
(1992 ),
pp. 443–469 .
MR
1189673 Zbl
0767.20013 article
Abstract
People
BibTeX
Two (synchronous, asynchronous, or non-deterministic asynchronous) automatic structures on a group \( G \) are “equivalent” if their union is a non-deterministic asynchronous automatic structure. We discuss this relation, giving a classification of structures up to equivalence for abelian groups and partial results in some other cases. We also discuss a “boundary” of an asynchronous automatic structure. We show that it is an invariant of the equivalence class of the structure, and describe other properties. We describe a “rehabilitated boundary” which yields \( S^{n-1} \) for any automatic structure on \( \mathbb{Z}^n \) .
@article {key1189673m,
AUTHOR = {Neumann, Walter D. and Shapiro, Michael},
TITLE = {Equivalent automatic structures and
their boundaries},
JOURNAL = {Int. J. Algebra Comput.},
FJOURNAL = {International Journal of Algebra and
Computation},
VOLUME = {2},
NUMBER = {4},
YEAR = {1992},
PAGES = {443--469},
DOI = {10.1142/S021819679200027X},
NOTE = {MR:1189673. Zbl:0767.20013.},
ISSN = {0218-1967},
}
W. D. Neumann and M. Shapiro :
“Automatic structures and boundaries for graphs of groups Int. J. Algebra Comput.
4 : 4
(1994 ),
pp. 591–616 .
MR
1313129 Zbl
0832.20051 ArXiv
math/9306205 article
Abstract
People
BibTeX
We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product bijects to the product of the sets of biautomatic structures on the vertex groups. The set of automatic structures is much richer. Indeed, it is dense in the infinite product of the sets of automatic structures of all conjugates of the vertex groups. We classify these structures by a class of labelled graphs which “mimic” the underlying graph of the graph of groups. Analogous statements hold for asynchronous automatic structures. We also discuss the boundaries of these structures.
@article {key1313129m,
AUTHOR = {Neumann, Walter D. and Shapiro, Michael},
TITLE = {Automatic structures and boundaries
for graphs of groups},
JOURNAL = {Int. J. Algebra Comput.},
FJOURNAL = {International Journal of Algebra and
Computation},
VOLUME = {4},
NUMBER = {4},
YEAR = {1994},
PAGES = {591--616},
DOI = {10.1142/S0218196794000178},
NOTE = {ArXiv:math/9306205. MR:1313129. Zbl:0832.20051.},
ISSN = {0218-1967},
}
W. D. Neumann and M. Shapiro :
“Automatic structures, rational growth, and geometrically finite hyperbolic groups Invent. Math.
120 : 2
(1995 ),
pp. 259–287 .
MR
1329042 Zbl
0831.20041 ArXiv
math/9401201 article
Abstract
People
BibTeX
We show that the set \( \mathbf{S}\mathfrak{U}(G) \) of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group \( G \) is dense in the product of the sets \( \mathbf{S}\mathfrak{U}(P) \) over all maximal parabolic subgroups \( P \) . The set \( \mathbf{BS}\mathfrak{U}(G) \) of equivalence classes of biautomatic structures on \( G \) is isomorphic to the product of the sets \( \mathbf{BS}\mathfrak{U}(P) \) over the cusps (conjugacy classes of maximal parabolic subgroups) of \( G \) . Each maximal parabolic \( P \) is a virtually abelian group, so \( \mathbf{S}\mathfrak{U}(P) \) and \( \mathbf{BS}\mathfrak{U}(P) \) were computed in [Neumann and Shapiro 1992].
We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics for \( G \) is regular. Moreover, the growth function of \( G \) with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.
@article {key1329042m,
AUTHOR = {Neumann, Walter D. and Shapiro, Michael},
TITLE = {Automatic structures, rational growth,
and geometrically finite hyperbolic
groups},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {120},
NUMBER = {2},
YEAR = {1995},
PAGES = {259--287},
DOI = {10.1007/BF01241129},
NOTE = {ArXiv:math/9401201. MR:1329042. Zbl:0831.20041.},
ISSN = {0020-9910},
}
W. D. Neumann and M. Shapiro :
A short course in geometric group theory ,
1996 .
online notes.
Notes for the ANU Workshop January/February 1996.
misc
People
BibTeX
@misc {key40963537,
AUTHOR = {Neumann, Walter D. and Shapiro, Michael},
TITLE = {A short course in geometric group theory},
HOWPUBLISHED = {online notes},
YEAR = {1996},
PAGES = {35},
NOTE = {(Canberra, 22 January--9 February 1996).
Notes for the ANU Workshop January/February
1996.},
}
W. D. Neumann and M. Shapiro :
“Regular geodesic normal forms in virtually abelian groups Bull. Aust. Math. Soc.
55 : 3
(1997 ),
pp. 517–519 .
MR
1456281 Zbl
0887.20015 ArXiv
math/9702203 article
Abstract
People
BibTeX
@article {key1456281m,
AUTHOR = {Neumann, Walter D. and Shapiro, Michael},
TITLE = {Regular geodesic normal forms in virtually
abelian groups},
JOURNAL = {Bull. Aust. Math. Soc.},
FJOURNAL = {Bulletin of the Australian Mathematical
Society},
VOLUME = {55},
NUMBER = {3},
YEAR = {1997},
PAGES = {517--519},
DOI = {10.1017/S0004972700034171},
NOTE = {ArXiv:math/9702203. MR:1456281. Zbl:0887.20015.},
ISSN = {0004-9727},
}
Geometric group theory down under Canberra, 14–19 July 1996 ).
Edited by J. Cossey, C. F. Miller, III, W. D. Neumann, and M. Shapiro .
de Gruyter (Berlin ),
1999 .
MR
1714835 Zbl
0910.00040 book
People
BibTeX
@book {key1714835m,
TITLE = {Geometric group theory down under},
EDITOR = {Cossey, John and Miller, III, Charles
F. and Neumann, Walter D. and Shapiro,
Michael},
PUBLISHER = {de Gruyter},
ADDRESS = {Berlin},
YEAR = {1999},
PAGES = {xii+333},
DOI = {10.1515/9783110806861},
NOTE = {(Canberra, 14--19 July 1996). MR:1714835.
Zbl:0910.00040.},
ISBN = {9783110163667},
}
C. F. Miller, III, W. D. Neumann, and G. A. Swarup :
“Some examples of hyperbolic groups 195–202
in
Geometric group theory down under
(Canberra, 14–19 July 1996 ).
Edited by J. Cossey, C. F. Miller, III, W. D. Neumann, and M. Shapiro .
de Gruyter (Berlin ),
1999 .
MR
1714846 Zbl
0955.20027 incollection
Abstract
People
BibTeX
@incollection {key1714846m,
AUTHOR = {Miller, III, C. F. and Neumann, Walter
D. and Swarup, G. A.},
TITLE = {Some examples of hyperbolic groups},
BOOKTITLE = {Geometric group theory down under},
EDITOR = {Cossey, John and Miller, III, Charles
F. and Neumann, Walter D. and Shapiro,
Michael},
PUBLISHER = {de Gruyter},
ADDRESS = {Berlin},
YEAR = {1999},
PAGES = {195--202},
DOI = {10.1515/9783110806861.195},
NOTE = {(Canberra, 14--19 July 1996). MR:1714846.
Zbl:0955.20027.},
ISBN = {9783110163667},
}
W. D. Neumann and M. Shapiro :
“Automatic structures on central extensions 261–280
in
Geometric group theory down under
(Canberra, 14–19 July 1996 ).
Edited by J. Cossey, C. F. Miller, III, W. D. Neumann, and M. Shapiro .
de Gruyter (Berlin ),
1999 .
MR
1714849 Zbl
1114.20305 incollection
Abstract
People
BibTeX
We show that a central extension of a group \( H \) by an abelian group \( A \) has an automatic structure with \( A \) a rational subgroup if and only if \( H \) has an automatic structure for which the extension is given by a “regular” cocycle. This had been proved for biautomatic structures by Neumann and Reeves. We make a start at classifying automatic structures on such groups, but we show that, at least for automatic structures, a classication using “controlling subgroups,” as done by the authors in certain other cases, is impossible.
@incollection {key1714849m,
AUTHOR = {Neumann, Walter D. and Shapiro, Michael},
TITLE = {Automatic structures on central extensions},
BOOKTITLE = {Geometric group theory down under},
EDITOR = {Cossey, John and Miller, III, Charles
F. and Neumann, Walter D. and Shapiro,
Michael},
PUBLISHER = {de Gruyter},
ADDRESS = {Berlin},
YEAR = {1999},
PAGES = {261--280},
DOI = {0.1515/9783110806861.261},
NOTE = {(Canberra, 14--19 July 1996). MR:1714849.
Zbl:1114.20305.},
ISBN = {9783110163667},
}
