## Isoperimetric and isodiametric functions of finite presentations

Venue: | in Geometric Group Theory |

Citations: | 27 - 3 self |

### BibTeX

@INPROCEEDINGS{Gersten_isoperimetricand,

author = {S. M. Gersten},

title = {Isoperimetric and isodiametric functions of finite presentations},

booktitle = {in Geometric Group Theory},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We survey current work relating to isoperimetric functions and isodiametric functions of finite presentations. §1. Introduction and Definitions Isoperimetric functions are classical in differential geometry, but their use in group theory derives from Gromov’s seminal article [Gr] and his characterization of word hyperbolic groups by a linear isoperimetric inequality. Isodiametric functions were introduced in our article [G1] in an attempt to provide a group theoretic

### Citations

424 |
Combinatorial group theory
- Lyndon, Schupp
- 1977
(Show Context)
Citation Context ... ɛi = ±1, ui ∈ F and ℓ(ui) ≤ f(ℓ(w)) + M. Here we write a b = bab −1 for elements a and b in a group. A word of caution is necessary here. A diagram of minimal area is always reduced, in the sense of =-=[LS]-=-. However this will not be the case in general for a diagram of minimal diameter. This complicates considerably the problem of proving that a diagram is diametrically minimal. Consequently we do not i... |

335 |
Hyperbolic groups, Essays in Group Theory
- Gromov
- 1987
(Show Context)
Citation Context ...functions of finite presentations. §1. Introduction and Definitions Isoperimetric functions are classical in differential geometry, but their use in group theory derives from Gromov’s seminal article =-=[Gr]-=- and his characterization of word hyperbolic groups by a linear isoperimetric inequality. Isodiametric functions were introduced in our article [G1] in an attempt to provide a group theoretic framewor... |

296 | Three dimensional manifolds, Kleinian groups and hyperbolic geometry
- Thurston
- 1982
(Show Context)
Citation Context .... 3.5.1. Lattices in the 3-dimensional Lie group Nil. 3.5.2. Lattices in the 3-dimensional Lie group Sol. 3.5.3. π1(M), where M is a compact 3-manifolds for which Thurston’s geometrization conjecture =-=[Th]-=- holds. The statements about lattices in Nil and Sol are proved in [G1]. Here is an extremely rough sketch for lattices in Sol. The ( problem ) is reduced to showing that 2 1 1 the “Fibonacci group” φ... |

216 | Structures métriques pour les variétés riemanniennes, Cedic-Fernand - Gromov, Lafontaine, et al. - 1981 |

205 |
Asymptotic invariants of infinite groups
- Gromov
- 1991
(Show Context)
Citation Context ...rimetric inequality (see also Section 5 below). Furthermore, it is shown in [ECHLPT] that an isoperimetric function for Sl3( ) must grow at least exponentially. Compare also the arguments sketched in =-=[Gr2]-=-. Proposition 3.4. If the group G is combable, then it satisfies the linear isodiametric inequality. Proof. Let σ : G → A ⋆ be a combing, where A is a finite set of semigroup generators. Suppose σ sat... |

199 | Introduction to Algebraic K-theory - Milnor - 1971 |

175 | Combinatorial group theory - Magnus, Karras, et al. - 1976 |

112 |
Topology of finite graphs
- Stallings
- 1983
(Show Context)
Citation Context ...pply this algorithm to decide whether or not w ∈ Nm. This solves the word problem for G. A particularly attractive geometric way of deciding whether or not w ∈ Nm as above has been given by Stallings =-=[St1]-=-. His algorithm amounts to using an immersion of finite graphs as a finite state automaton. Note that the automaton depends on the word w being tested. Remark. It is somewhat mysterious that one has t... |

107 |
A combination theorem for negatively curved groups
- Bestvina, Feighn
- 1992
(Show Context)
Citation Context ...ity. Question. If φ is an automorphism of the finitely generated free group F and if G = F φ is the corresponding split extension, does G satisfy the quadratic isoperimetric inequality? The result of =-=[BF]-=-, that G is word hyperbolic if and only if it contains no subgroup isomorphic to 2 , can be viewed as positive evidence. Furthermore, G is automatic if φ is geometric (that is, if φ is induced by a ho... |

53 |
Automatic groups and amalgams
- Baumslag, Gersten, et al.
- 1991
(Show Context)
Citation Context ...chronously combable group is finitely presented and satisfies a linear isodiametric inequality. Furthermore, if G is asynchronously automatic, then it has an exponential isoperimetric function[ECHLPT]=-=[BGSS]-=-. Remark. In the original version of this survey, we raised the question whether the integral Heisenberg group was combable. Gromov asserted at the conference that the real Heisenberg group is combabl... |

31 |
In¶egalit¶es isop¶erim¶etriques et quasi-isom¶etrties
- Alonso
- 1991
(Show Context)
Citation Context ... of proving that a diagram is diametrically minimal. Consequently we do not introduce a diametric analog of the Dehn function. Next we discuss the question of change of presentation. Proposition 1.1, =-=[Al]-=-[Sh]. Let P and P ′ be finite presentations for isomorphic groups. If f is an isoperimetric function (resp. isodiametric function) for P, then there exist positive constants A, B, C, D, and E such tha... |

18 |
Ol’shanskii, Hyperbolicity of groups with subquadratic isoperimetric inequality
- Yu
- 1991
(Show Context)
Citation Context ...nite presentation with a subquadratic isoperimetric function also possesses a linear isoperimetric function [Gr, 2.3.F]. A. Yu. Ol’shanskii recently found an elementary proof of this important result =-=[Ol]-=-.6 ISOPERIMETRIC In order to explain how quadratic isoperimetric functions arise, it is necessary to introduce new notions. Definition 3.2. Let G be a finitely generated group with finite set A of se... |

15 |
A presentation for the automorphism group of a free group of finite rank
- McCool
- 1974
(Show Context)
Citation Context ...ngth of a cyclically reduced word conjugate to w. The generators for Aut(F ) are taken to be the Whitehead automorphisms [LS]. The function fµ is seen to grow exponentially with n. McCool’s algorithm =-=[Mc]-=- is a peak reduction algorithm for these data, so the assertion for Aut(F ) follows from Theorem 4.1. The argument for Out(F ) is similar. Remark. These results for Aut(F ) and Out(F ) are surely not ... |

12 |
Casson’s idea about 3-manifolds whose universal cover is R 3 , Intern
- Gersten, Stallings
- 1991
(Show Context)
Citation Context ...fies condition ID(α), where α > 0, if there is an ɛ ≥ 0 so that n ↦→ αn + ɛ is an isodiametric function for P. This is of course just a reformulation of a linear isodiametric inequality. Theorem 3.6, =-=[SG]-=-. Let M be a closed, orientable, irreducible, aspherical 3manifold whose fundamental group admits a finite presentation satisfying condition ID(α), where α < 1. Then the universal cover of M is homeom... |

9 |
The double exponential theorem for isodiametric and isoperimetric functions
- Gersten
- 1991
(Show Context)
Citation Context ...nstants a, b so that n ↦→ abf(n)+n is an isoperimetric function for P. His proof makes use of an analysis of Nielsen’s reduction process for producing a basis for a subgroup of a free group (see also =-=[G4]-=- for a different treatment involving Stallings’ folds). Here is a striking example, which shows that the complexity of the word problem for 1-relator groups, as measured by the growth of an isodiametr... |

8 |
Bounded cohomology and combings of groups, preprint, Univ. of Utah, available at ftp://ftp.math.utah.edu/u/ma/gersten/bdd.dvi.Z
- Gersten
(Show Context)
Citation Context ...on’s result (Theorem 3.3) is quoted for the proof. However, this last result applies only for a linearly bounded combing, so it must be considered open whether Sl3( ) is combable or not. We proved in =-=[G3]-=- that all combable groups satisfy an exponential isoperimetric inequality, and this is the best result known to date in this generality.8 ISOPERIMETRIC Remark. There is an analogous notion of asynchr... |

7 |
Isodiametric and isoperimetric inequalities for group presentations
- Cohen
- 1991
(Show Context)
Citation Context ...ould be an isoperimetric function of the form n ↦→ af(n)+n , for a constant a (Stallings raised this question in the special case when f is linear). In this connection, D. E. Cohen has recently shown =-=[C]-=- that if f is an isodiametric function for a finite presentation P, then there are positive constants a, b so that n ↦→ abf(n)+n is an isoperimetric function for P. His proof makes use of an analysis ... |

7 |
Isodiametric and isoperimetric inequalities of group extensions
- Gersten
- 1991
(Show Context)
Citation Context ... group theory derives from Gromov’s seminal article [Gr] and his characterization of word hyperbolic groups by a linear isoperimetric inequality. Isodiametric functions were introduced in our article =-=[G1]-=- in an attempt to provide a group theoretic framework for a result of Casson’s (see Theorem 3.6 below). It turned out subsequently that the notion had been considered earlier under a different name [F... |

7 |
Geometry à la Gromov for the fundamental group of a closed 3-manifold M 3 and the simple connectivity at infinity
- Poénaru
- 1994
(Show Context)
Citation Context ...]. The difficulty with these conditions of 4 course is that they are not invariant under change of generators.10 ISOPERIMETRIC We should also cite additional work in connection with Casson’s theorem =-=[P]-=-[Br][St2]. §4. Relation with Peak Reduction Algorithms In this section, G will denote a finitely presented group with finite set of semigroup generators A and associated Cayley graph Γ. Definition 4.1... |

6 |
Die Gruppe der dreidimensionalen Gittertransformationen, Danske Vid
- Nielsen
(Show Context)
Citation Context ...n for Out(F ) when rank(F ) = 3 is still open. Theorem 4.5. Sl3( ) has an exponential isodiametric function and an isoperimetric function of the form n ↦→ ABn. This follows from a result of Nielsen’s =-=[N]-=-, that the group Sl3( ) satisfies a peak reduction algorithm for the function µ given by µ(x) = ( ∑ x2 ij ) − 3, for x ∈ Sl3( ). The generators here are the elementary transvections Eij(1) and the sig... |

6 |
A Geometric approach to the almost convexity and growth of some nilpotent groups
- Shapiro
- 1989
(Show Context)
Citation Context ... an almost convex Cayley graph, in the sense of Cannon [Ca], has an ID( 1 2 ) presentation [G1] (see also §4 below). Since it is known that every Nil group has at least one almost convex Cayley graph =-=[Sp1]-=-, it follows that Nil groups have ID( 1 2 ) presentations. Another argument, proving that Nil groups have ID( 3) presentations, appears in [G1]. The difficulty with these conditions of 4 course is tha... |

4 | The word problem for geometrically finite groups”, Geometriae Dedicata 20 - Floyd, Hoare, et al. - 1986 |

4 |
functions and ℓ1-norms of finite presentations, Algorithms and classification in combinatorial group theory
- Gersten, Dehn
- 1987
(Show Context)
Citation Context ...operimetric Functions The methods of this section for establishing lower bounds for the Dehn function of a finite presentation are due to [BMS] (other methods for finding lower bounds can be found in =-=[G2]-=-). We shall prove that the Dehn function of the free nilpotent group on p ≥ 2 generators of class c grows at least as fast as a polynomial of degree c + 1. Since it is known that every finitely genera... |

4 |
Deterministic and nondeterministic asynchronous automatic structures’. In: International Journal of Algebra and Computation 2.3, pp. 297– 305 (cit. on pp
- Shapiro
- 1992
(Show Context)
Citation Context ...c structure if the language σ(G) ⊂ A ⋆ is regular. M. Shapiro has recently proved that the definition just given is equivalent to that of [ECHLPT] for an asynchronously automatic structure on a group =-=[Sp2]-=-. One can show that every asynchronously combable group is finitely presented and satisfies a linear isodiametric inequality. Furthermore, if G is asynchronously automatic, then it has an exponential ... |

3 | Groups and combings
- Short
- 1990
(Show Context)
Citation Context ...proving that a diagram is diametrically minimal. Consequently we do not introduce a diametric analog of the Dehn function. Next we discuss the question of change of presentation. Proposition 1.1, [Al]=-=[Sh]-=-. Let P and P ′ be finite presentations for isomorphic groups. If f is an isoperimetric function (resp. isodiametric function) for P, then there exist positive constants A, B, C, D, and E such that n ... |

2 |
Almost convex groups. Geometriae Dedicata 22
- Cannon
- 1987
(Show Context)
Citation Context ... is such that each word σ(g) is geodesic has a finite presentation satisfying condition ID( 1 2 ). Every finitely presented group which possesses an almost convex Cayley graph, in the sense of Cannon =-=[Ca]-=-, has an ID( 1 2 ) presentation [G1] (see also §4 below). Since it is known that every Nil group has at least one almost convex Cayley graph [Sp1], it follows that Nil groups have ID( 1 2 ) presentati... |

1 |
Combing the fundamental group of a Haken 3-manifold
- Bridson
- 1992
(Show Context)
Citation Context ...cceeded in proving the weaker result that the group is asynchronously combable; so we regard this as an open question of great interest. In this connection, we mention a recent result of M. Bridson’s =-=[Bd]-=-, that the group n φ is asynchronously combable for all φ ∈ Gln( ). Theorem 3.5. The following finitely presented groups all have linear isodiametric functions. 3.5.1. Lattices in the 3-dimensional Li... |

1 |
Isoperimetric inequalities and the homology of groups, preprint
- Baumslag, Miller, et al.
- 1992
(Show Context)
Citation Context ...) have a linear isodiametric function? §5. Lower bounds for Isoperimetric Functions The methods of this section for establishing lower bounds for the Dehn function of a finite presentation are due to =-=[BMS]-=- (other methods for finding lower bounds can be found in [G2]). We shall prove that the Dehn function of the free nilpotent group on p ≥ 2 generators of class c grows at least as fast as a polynomial ... |

1 |
Filtrations of universal covers and a property of groups, preprint
- Brick
- 1991
(Show Context)
Citation Context ...The difficulty with these conditions of 4 course is that they are not invariant under change of generators.10 ISOPERIMETRIC We should also cite additional work in connection with Casson’s theorem [P]=-=[Br]-=-[St2]. §4. Relation with Peak Reduction Algorithms In this section, G will denote a finitely presented group with finite set of semigroup generators A and associated Cayley graph Γ. Definition 4.1. Le... |

1 |
Les groupes hyperboliques, Sem
- Ghys
- 1989
(Show Context)
Citation Context ...ition 1.1 below), isoperimetric and isodiametric functions are quasiisometry invariants of finitely presented groups. Hence these functions are examples of geometric properties, in the terminology of =-=[Gh]-=-. If P = 〈x1, x2, . . . , xp | R1, R2, . . . Rq〉 is a finite presentation, we shall denote by G = G(P) the associated group; here G = F/N, where F is the free group freely generated by the generators ... |

1 |
PRL for general linear groups
- Kalajd˘zievski
- 1992
(Show Context)
Citation Context ...free group freely generated by the generators x1, x2, . . . , xp and let N ⊳ F be the normal closure of the relators. We let G = G(P) = F/N as earlier. 1 We have in the meantime received the preprint =-=[Ka]-=- which contains the peak reduction lemma for the general linear groups.12 ISOPERIMETRIC Proposition 5.1. The group N/[F, N] is a finitely generated abelian group. Proof. The identity R u i = [u, Ri]R... |

1 |
Brick’s “QSF”: a group-theoretic idea related to simple connectivity at infinity of 3-manifolds, to appear
- Stallings
- 1991
(Show Context)
Citation Context ...difficulty with these conditions of 4 course is that they are not invariant under change of generators.10 ISOPERIMETRIC We should also cite additional work in connection with Casson’s theorem [P][Br]=-=[St2]-=-. §4. Relation with Peak Reduction Algorithms In this section, G will denote a finitely presented group with finite set of semigroup generators A and associated Cayley graph Γ. Definition 4.1. Let µ :... |