Fractional isoperimetric inequalities and subgroup distortion

Bridson, Martin

doi:10.1090/S0894-0347-99-00308-2

Fractional isoperimetric inequalities and subgroup distortion
HTML articles powered by AMS MathViewer

by Martin R. Bridson PDF

J. Amer. Math. Soc. 12 (1999), 1103-1118 Request permission

Abstract:

It is shown that there exist infinitely many non-integers $r>2$ such that the Dehn function of some finitely presented group is $\simeq n^r$. Explicit examples of such groups are constructed. For each rational number $s\ge 1$ pairs of finitely presented groups $H\subset G$ are constructed so that the distortion of $H$ in $G$ is $\simeq n^s$.

References

Juan M. Alonso, Inégalités isopérimétriques et quasi-isométries, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 12, 761–764 (French, with English summary). MR 1082628
Martin R. Bridson, On the geometry of normal forms in discrete groups, Proc. London Math. Soc. (3) 67 (1993), no. 3, 596–616. MR 1238046, DOI 10.1112/plms/s3-67.3.596
M. R. Bridson, Combings of semidirect products and $3$-manifold groups, Geom. Funct. Anal. 3 (1993), no. 3, 263–278. MR 1215781, DOI 10.1007/BF01895689
Martin R. Bridson, Optimal isoperimetric inequalities for abelian-by-free groups, Topology 34 (1995), no. 3, 547–564. MR 1341808, DOI 10.1016/0040-9383(94)00044-L
M. R. Bridson, Asymptotic cones and polynomial isoperimetric inequalities, Topology 38 (1999), 543–554.
M. R. Bridson, Area versus diameter in van Kampen diagrams, in preparation.
N. Brady and M. R. Bridson, There is only one gap in the isoperimetric spectrum, preprint, University of Oxford, 1998.
M. R. Bridson and S. M. Gersten, The optimal isoperimetric inequality for torus bundles over the circle, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 1–23. MR 1380947, DOI 10.1093/qmath/47.1.1
G. Baumslag, C. F. Miller III, and H. Short, Isoperimetric inequalities and the homology of groups, Invent. Math. 113 (1993), no. 3, 531–560. MR 1231836, DOI 10.1007/BF01244317
B. H. Bowditch, A short proof that a subquadratic isoperimetric inequality implies a linear one, Michigan Math. J. 42 (1995), no. 1, 103–107. MR 1322192, DOI 10.1307/mmj/1029005156
M. R. Bridson and Ch. Pittet, Isoperimetric inequalities for the fundamental groups of torus bundles over the circle, Geom. Dedicata 49 (1994), no. 2, 203–219. MR 1266274, DOI 10.1007/BF01610621
Steve M. Gersten, Isoperimetric and isodiametric functions of finite presentations, Geometric group theory, Vol. 1 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 181, Cambridge Univ. Press, Cambridge, 1993, pp. 79–96. MR 1238517, DOI 10.1017/CBO9780511661860.008
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
John R. Stallings and S. M. Gersten, Casson’s idea about $3$-manifolds whose universal cover is $\textbf {R}^3$, Internat. J. Algebra Comput. 1 (1991), no. 4, 395–406. MR 1154440, DOI 10.1142/S0218196791000274
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
John Mitchell, On Carnot-Carathéodory metrics, J. Differential Geom. 21 (1985), no. 1, 35–45. MR 806700
Graham A. Niblo and Martin A. Roller (eds.), Geometric group theory. Vol. 1, London Mathematical Society Lecture Note Series, vol. 181, Cambridge University Press, Cambridge, 1993. MR 1238510
A. Yu. Ol′shanskiĭ, Hyperbolicity of groups with subquadratic isoperimetric inequality, Internat. J. Algebra Comput. 1 (1991), no. 3, 281–289. MR 1148230, DOI 10.1142/S0218196791000183
A. Yu. Ol′shanskiĭ, On the distortion of subgroups of finitely presented groups, Mat. Sb. 188 (1997), no. 11, 51–98 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 11, 1617–1664. MR 1601512, DOI 10.1070/SM1997v188n11ABEH000276
A. Yu. Ol’shanskii and M. V. Sapir, Length and area functions on groups and quasi-isometric Higman embeddings, Internat. J. Algebra Comput., to appear.
Panagiotis Papasoglu, On the sub-quadratic isoperimetric inequality, Geometric group theory (Columbus, OH, 1992) Ohio State Univ. Math. Res. Inst. Publ., vol. 3, de Gruyter, Berlin, 1995, pp. 149–157. MR 1355109
Ch. Pittet, Isoperimetric inequalities for homogeneous nilpotent groups, Geometric group theory (Columbus, OH, 1992) Ohio State Univ. Math. Res. Inst. Publ., vol. 3, de Gruyter, Berlin, 1995, pp. 159–164. MR 1355110
M. Sapir, J-C. Birget and E. Rips, Isoperimetric and isodiametric functions of groups, preprint, Vanderbilt University, 1997.
Ruth Charney, Michael Davis, and Michael Shapiro (eds.), Geometric group theory, Ohio State University Mathematical Research Institute Publications, vol. 3, Walter de Gruyter & Co., Berlin, 1995. MR 1355100, DOI 10.1515/9783110810820