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Measuring the tameness of almost convex groups

Author(s): Susan Hermiller; John Meier
Journal: Trans. Amer. Math. Soc. 353 (2001), 943-962.
MSC (2000): Primary 20F65; Secondary 20F69, 57M07
Posted: October 11, 2000
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Abstract:

A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points $x$ in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi-isometry invariants. We show that a group is almost convex if and only if the radial tameness function is bounded by the identity function; hence almost convex groups, as well as certain generalizations of almost convex groups, are contained in the quasi-isometry class of groups admitting linear radial tameness functions.

References:

[1]
S.G. Brick, Quasi-isometries and amalgamations of tame combable groups, Internat. J. Comput. Algebra 5 (1995), 199-204. MR 96c:20064

[2]
J.W. Cannon, Almost convex groups, Geom. Dedicata 22 (1987), 197-210. MR 88a:20049

[3]
D.B.A. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, and W.P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992. MR 93i:20036

[4]
S.M. Gersten, Dehn functions and $l_{1}$-norms of finite presentations, In: Algorithms and classification in combinatorial group theory, G. Baumslag, C.F. Miller III, eds., (M.S.R.I. Publications), Springer 23 (1992), 195-224. MR 94g:20049

[5]
S.M. Gersten, Isoperimetric and isodiametric functions of finite presentations, In: Geometric group theory, vol. 1, G.A. Niblo and M.A. Roller, eds., (London Math. Soc. Lecture Note Ser.) Cambridge: Cambridge University Press 181 (1993), 79-96. MR 94f:20066

[6]
S.M. Hermiller and J. Meier, Tame combings, almost convexity, and rewriting systems for groups, Math. Z. 225 (1997), 263-276. MR 98i:20036

[7]
C. Hog-Angeloni, W. Metzler and A.J. Sieradski, eds., Two-dimensional Homotopy and Combinatorial Group Theory. London Math. Soc. Lecture Notes Series 197, Cambridge University Press, Cambridge, 1993. MR 95g:57006

[8]
D.E. Knuth and P.B. Bendix, Simple word problems in universal algebras, In: Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) (1970), 263-297. MR 41:134

[9]
R.C. Lyndon and P.E. Schupp, Combinatorial group theory, (Ergeb. Math., Bd. 89) Springer, Berlin-Heidelberg-New York, 1977. MR 58:28182

[10]
M. Machtey and P. Young, An introduction to the general theory of algorithms, North-Holland, New York, 1978. MR 81k:68001

[11]
M.L. Mihalik, Compactifying coverings of $3$-manifolds, Comment. Math. Helv. 71 (1996), 362-372. MR 97k:57020

[12]
M.L. Mihalik, Group extensions and tame pairs, Trans. Amer. Math. Soc. 351 (1999), 1095-1107. MR 99e:57002

[13]
M.L. Mihalik and S.T. Tschantz, Tame combings of groups, Trans. Amer. Math. Soc. 349 (1997), 4251-4264. MR 97m:20049

[14]
V. Poénaru, Almost convex groups, Lipschitz combing, and $\pi ^{\infty }_{1}$ for universal covering spaces of closed 3-manifolds, J. Differential Geom. 35 (1992), 103-130. MR 93d:57032

[15]
V. Poénaru, Geometry ``à la Gromov'' for the fundamental group of a closed 3-manifold $M^{3}$ and the simple connectivity at infinity of $\widetilde {M}^{3}$, Topology 33 (1994), 181-196. MR 94m:57034

[16]
V. Poénaru and C. Tanasi, k-weakly almost convex groups and $\pi ^{\infty }_{1} \widetilde {M}^{3}$, Geom. Dedicata 48 (1993), 57-81. MR 94k:57003

[17]
C.C. Sims, Computation with finitely presented groups. (Encyclopedia of Mathematics and its Applications 48), Cambridge University Press, Cambridge, 1994. MR 95f:20053

[18]
C. Thiel, Zur Fast-Konvexität einiger nilpotenter Gruppen, Bonner Mathematische Schriften (1992). MR 95e:20052

[19]
T. Tucker, Non-compact 3-manifolds and the missing boundary problem, Topology 13 (1974), 267-273. MR 50:5801

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Additional Information:

Susan Hermiller
Affiliation: Department of Mathematics and Statistics, University of Nebraska, Lincoln, Nebraska 68588-0323
Email: smh@math.unl.edu

John Meier
Affiliation: Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042
Email: meierj@lafayette.edu

DOI: 10.1090/S0002-9947-00-02717-3
PII: S 0002-9947(00)02717-3
Keywords: Tame combings, almost convex groups, covering conjecture, rewriting systems, isoperimetric inequalities
Received by editor(s): June 21, 1999
Posted: October 11, 2000
Additional Notes: Susan Hermiller acknowledges support from NSF grant DMS-9623088
John Meier acknowledges support from NSF RUI grant DMS-9704417
Copyright of article: Copyright 2000, American Mathematical Society

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