Group theory Research
Selected older E-prints:
Two of the following papers were published in a journal
that is hard to find in libraries. If you can't find something
here, write me and I'll try to find the file.
- Dehn functions and L_1-norms of finite presentions,
.pdf,
appeared in
"Algorithms and Classification in
Combinatorial Group Theory", edited by G.
Baumslag and C. F. Miller III, Springer-Verlag,
MSRI series vol. 23,
1991.
In this paper I applied techniques
of algebraic topology to compute lower bounds for Dehn functions
(Gromov calls one of these techniques "Gersten's lemma", which
is used but not stated explicitly in this paper), and I
produced an example of a 1-relator group whose Dehn function, as
I prove here,
grows faster than any iterated exponential.
The figures (done in an obsolete version of MacDraw) are not
available in this file.
- (with H. Short) Rational subgroups of biautomatic groups,
.pdf,
appeared in
Annals of Mathematics, vol. 134 (1991) 125--158.
-
Bounded cocycles and combings of groups,
.dvi,
.ps,
.pdf,
.tex
appeared in
International Journal of Algebra and Computation 2 (1992)
307-326 (with an appendix by Domingo Toledo).
The main technical result of this paper
(Theorem 3.1) is that a central infinite cyclic
extension of a finitely generated group G defined by a bounded cocycle is
quasi-isometric to the product Z x G.
The result has two important applications, to the coincidence of two of the
8 geometries of 3-manifolds up to quasi-isometry and to
the existence of a combable group which is not residually
finite, both of which were first proved in this paper.
- The double exponential theorem for isodiametric and isoperimetric
functions
.ps,
.pdf,
.dvi,
appeared in International Journal of Algebra and Computation 1(1991)
321-327. This paper contains a topological proof, based on the
notion of folding due to Stallings, of the theorem first
proved by D. E. Cohen, by a complexity analysis of
the Nielsen reduction algorithm, that for every finitely
presented group there is an isoperimetric function which is a double exponential
in its minimal isodiametric function.
-
Isoperimetric and isodiametric functions of finite presentations
.ps,
.pdf,
.dvi,
appeared in Geometric Group Theory, Volume 1, edited by G. Niblo and
M Roller, London Math. Society Lecture Notes Series
181 (1993) 79-96, Cambridge Univ. Press.
The Dehn function and isodiametric functions of finite presentations
are introduced. The word problem is solvable iff the Dehn function
is recursive.
- Finiteness properties of asynchronously automatic groups
.dvi,
appeared in ``Geometric Group Theory", Proceedings of a Special Research Quarter
at The Ohio state University, Spring 1992, edited
by Ruth Charney, Michael Davis, and Michael Shapiro,
pp. 121--134, Ohio State Univ. Mathematical
Research Institute Publ. vol. 3, de Gruyter 1995.
Several results related to asynchronous combings are proved in
this paper. The two most significant, I now believe, are
(1) all asynchronously combable groups have linearly bounded
filling length, and (2) asynchronously automatic groups
are of type F_3. The first is of interest since all compact
3-manifold groups satisfying Thurston's geometrization
conjecture are asynchronously combable. (The property of linearly bounded filling
length is called LCNH_1 in this paper; it was written a decade before
Riley and I worked out the general theory of filling length of finitely
presented groups.)
- Preservation and distortion of area in finitely presented groups
.ps,
.pdf,
.dvi,
appeared in GAFA, 6 (1996) 301--345.
- Cohomological
lower bounds for isoperimetric functions on groups
.ps,
.pdf,
.dvi
appeared
in Topology, 37 no. 5 (1998) pp. 1031-1072 (the
publisher requires that, in order to
the preprint version on my homepage,
I cannot make any changes from the final published version.)
The converse for the combination theorem for hyperbolic
groups appears here.
- Banff lectures
on hyperbolic and automatic groups
.ps,
.pdf,
.dvi,
appeared in the Proceedings of the CRM Summer School in Combinatorial
Group Theory, Banff 1996, editor O. Kharlampovich, published by the AMS
in the CRM Proceedings and Lecture Notes Series, vol. 17 (1999)
(the web version incorporates corrections).
This is the written version of two introductory lectures I gave
at Banff. Some proofs are given while many are referred to the
literature. There is a complete discussion of small cancellation
conditions and hyperbolicity based on metric weight tests which
is due to me and for which there is no other source in the literature.
- A cohomological
characterization of hyperbolic groups
.pdf(Dec. 1996).
I never published this paper and it seems that I forgot about it.
I. Mineyev didn't, however, and he told me (2004) that he wanted to
refer to a result in it, so I'm listing it here.
- (with
D. Allcock) A homological character
ization of
hyperbolic groups
.ps,
.pdf,
.dvi,
appeared in Inventiones Math. 135
(1999) (3) 723-742.
This contains an approximation theorem of independent
interest, that every ell-one ($\ell_1$-) real-valued cycle on an arbitrary graph
can be approximated by cycles of compact support.
- Distortion and L1-homology
.ps,
.pdf,
.dvi,
appeared in Groups Korea 98, editors Baik, Johnson, and Kim,
pages 133-164, published by de Gruyter, 2000.