Dehn function

In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the ''area'' of a ''relation'' in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of the length of that relation (see pp. 79–80 in ). The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive (see Theorem 2.1 in ). The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface.

History

The idea of an isoperimetric function for a finitely presented group goes back to the work of Max Dehn in 1910s. Dehn proved that the word problem for the standard presentation of the fundamental group of a closed oriented surface of genus at least two is solvable by what is now called Dehn's algorithm. A direct consequence of this fact is that for this presentation the Dehn function satisfies Dehn(''n'') ≤ ''n''. This result was extended in 1960s by Martin Greendlinger to finitely presented groups satisfying the C'(1/6) small cancellation condition. The formal notion of an isoperimetric function and a Dehn function as it is used today appeared in late 1980s – early 1990s together with the introduction and development of the theory of word-hyperbolic groups. In his 1987 monograph "Hyperbolic groups"M. Gromov, ''Hyperbolic Groups'' in: "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75–263. . Gromov proved that a finitely presented group is word-hyperbolic if and only if it satisfies a linear isoperimetric inequality, that is, if and only if the Dehn function of this group is equivalent to the function ''f''(''n'') = ''n''. Gromov's proof was in large part informed by analogy with filling area functions for compact Riemannian manifolds where the area of a minimal surface bounding a null-homotopic closed curve is bounded in terms of the length of that curve.

The study of isoperimetric and Dehn functions quickly developed into a separate major theme in geometric group theory, especially since the growth types of these functions are natural quasi-isometry invariants of finitely presented groups. One of the major results in the subject was obtained by Sapir, Birget and Rips who showed that most "reasonable" time complexity functions of Turing machines can be realized, up to natural equivalence, as Dehn functions of finitely presented groups.

Formal definition

Let
:$G=\langle X, R\rangle\qquad (*)$
be a finite group presentation where the ''X'' is a finite alphabet and where ''R'' ⊆ ''F''(''X'') is a finite set of cyclically reduced words.

Area of a relation

Let ''w'' ∈ ''F''(''X'') be a ''relation'' in ''G'', that is, a freely reduced word such that ''w'' = 1 in ''G''. Note that this is equivalent to saying that ''w'' belongs to the normal closure of ''R'' in ''F''(''X''), that is, there exists a representation of ''w'' as

:$w=u_1r_1u_1^\cdots u_m r_mu_^ \text F(X),$   (♠)

where ''m'' ≥ 0 and where ''r_i'' ∈ ''R''^{± 1} for ''i'' = 1, ..., ''m''.

For ''w'' ∈ ''F''(''X'') satisfying ''w'' = 1 in ''G'', the ''area'' of ''w'' with respect to (∗), denoted Area(''w''), is the smallest ''m'' ≥ 0 such that there exists a representation (♠) for ''w'' as the product in ''F''(''X'') of ''m'' conjugates of elements of ''R''^{± 1}.

A freely reduced word ''w'' ∈ ''F''(''X'') satisfies ''w'' = 1 in ''G'' if and only if the loop labeled by ''w'' in the presentation complex for ''G'' corresponding to (∗) is null-homotopic. This fact can be used to show that Area(''w'') is the smallest number of 2-cells in a van Kampen diagram over (∗) with boundary cycle labelled by ''w''.

Isoperimetric function

An ''isoperimetric function'' for a finite presentation (∗) is a monotone non-decreasing function
:$f: \mathbb N\to [0,\infty)$
such that whenever ''w'' ∈ ''F''(''X'') is a freely reduced word satisfying ''w'' = 1 in ''G'', then
:Area(''w'') ≤ ''f''(, ''w'', ),
where , ''w'', is the length of the word ''w''.

Dehn function

Then the ''Dehn function'' of a finite presentation (∗) is defined as

:$(n)=\max\$

Equivalently, Dehn(''n'') is the smallest isoperimetric function for (∗), that is, Dehn(''n'') is an isoperimetric function for (∗) and for any other isoperimetric function ''f''(''n'') we have
:Dehn(''n'') ≤ ''f''(''n'')
for every ''n'' ≥ 0.

Growth types of functions

Because the exact Dehn function usually depends on the presentation, one usually studies its asymptotic growth type as ''n'' tends to infinity, which only depends on the group.

For two monotone-nondecreasing functions
:$f,g: \mathbb N\to [0,\infty)$
one says that ''f'' is ''dominated'' by ''g'' if there exists ''C'' ≥1 such that
:$f(n)\le Cg(Cn+C)+Cn+C$
for every integer ''n'' ≥ 0. Say that ''f'' ≈ ''g'' if ''f'' is dominated by ''g'' and ''g'' is dominated by ''f''. Then ≈ is an equivalence relation and Dehn functions and isoperimetric functions are usually studied up to this equivalence relation.
Thus for any ''a,b > 1'' we have ''a''^''n'' ≈ ''b''^''n''. Similarly, if ''f''(''n'') is a polynomial of degree ''d'' (where ''d'' ≥ 1 is a real number) with non-negative coefficients, then ''f''(''n'') ≈ ''n''^''d''. Also, 1 ≈ ''n''.

If a finite group presentation admits an isoperimetric function ''f''(''n'') that is equivalent to a linear (respectively, quadratic, cubic, polynomial, exponential, etc.) function in ''n'', the presentation is said to satisfy a linear (respectively, quadratic, cubic, polynomial, exponential, etc.) ''isoperimetric inequality''.

Basic properties

*If ''G'' and ''H'' are quasi-isometric finitely presented groups and some finite presentation of ''G'' has an isoperimetric function ''f''(''n'') then for any finite presentation of ''H'' there is an isoperimetric function equivalent to ''f''(''n''). In particular, this fact holds for ''G'' = ''H'', where the same group is given by two different finite presentations.
*Consequently, for a finitely presented group the growth type of its Dehn function, in the sense of the above definition, does not depend on the choice of a finite presentation for that group. More generally, if two finitely presented groups are quasi-isometric then their Dehn functions are equivalent.
*For a finitely presented group ''G'' given by a finite presentation (∗) the following conditions are equivalent:
**''G'' has a recursive Dehn function with respect to (∗).
**There exists a recursive isoperimetric function ''f''(''n'') for (∗).
**The group ''G'' has solvable word problem.
::In particular, this implies that solvability of the word problem is a quasi-isometry invariant for finitely presented groups.
*Knowing the area Area(''w'') of a relation ''w'' allows to bound, in terms of , ''w'', , not only the number of conjugates of the defining relations in (♠) but the lengths of the conjugating elements ''u''_''i'' as well. As a consequence, it is knownS. M. Gersten
''Isoperimetric and isodiametric functions of finite presentations.''
Geometric group theory, Vol. 1 (Sussex, 1991), pp. 79–96, London Math. Soc. Lecture Note Ser., 181, Cambridge University Press, Cambridge, 1993. that if a finitely presented group ''G'' given by a finite presentation (∗) has computable Dehn function Dehn(''n''), then the word problem for ''G'' is solvable with non-deterministic time complexity Dehn(''n'') and deterministic time complexity Exp(Dehn(''n'')). However, in general there is no reasonable bound on the Dehn function of a finitely presented group in terms of the deterministic time complexity of the word problem and the gap between the two functions can be quite large.

Examples

*For any finite presentation of a finite group ''G'' we have Dehn(''n'') ≈ ''n''.Martin R. Bridson
''The geometry of the word problem.''
Invitations to geometry and topology, pp. 29–91, Oxford Graduate Texts in Mathematics, 7, Oxford University Press, Oxford, 2002. .
*For the closed oriented surface of genus 2, the standard presentation of its fundamental group
:G=\langle a_1,a_2,b_1,b_2, _1,b_1a_2,b_2]=1\rangle
:satisfies Dehn(''n'') ≤ ''n'' and Dehn(''n'') ≈ ''n''.
*For every integer ''k'' ≥ 2 the free abelian group $\mathbb Z^k$ has Dehn(''n'') ≈ ''n''².
*The Baumslag-Solitar group
:$B(1,2)=\langle a,b, b^ab=a^2\rangle$
:has Dehn(''n'') ≈ 2^''n'' (see ).
*The 3-dimensional discrete Heisenberg group
:H_3=\langle a,b, t, ,t ,t1, ,bt^2 \rangle
:satisfies a cubic but no quadratic isoperimetric inequality.
*Higher-dimensional Heisenberg groups
:H_=\langle a_1,b_1,\dots, a_k,b_k,t, _i,b_it, _i,t _i,t1, i=1,\dots, k, _i,b_j1, i\ne j\rangle,
:where ''k'' ≥ 2, satisfy quadratic isoperimetric inequalities.
*If ''G'' is a "Novikov-Boone group", that is, a finitely presented group with unsolvable word problem, then the Dehn function of ''G'' growths faster than any recursive function.
*For the Thompson group ''F'' the Dehn function is quadratic, that is, equivalent to ''n''² (see ).
*The so-called Baumslag-Gersten group
::$G=\langle a, t, (t^a^ t) a (t^ at)=a^2\rangle$
:has a Dehn function growing faster than any fixed iterated tower of exponentials. Specifically, for this group
::Dehn(''n'') ≈ exp(exp(exp(...(exp(1))...)))
:where the number of exponentials is equal to the integral part of log₂(''n'') (see ).

Known results

*A finitely presented group is word-hyperbolic group if and only if its Dehn function is equivalent to ''n'', that is, if and only if every finite presentation of this group satisfies a linear isoperimetric inequality.
*Isoperimetric gap: If a finitely presented group satisfies a subquadratic isoperimetric inequality then it is word-hyperbolic. Thus there are no finitely presented groups with Dehn functions equivalent to ''n''^''d'' with ''d'' ∈ (1,2).
*Automatic groups and, more generally, combable groups satisfy quadratic isoperimetric inequalities. D. B. A. Epstein, J. W. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston. '' Word Processing in Groups.'' Jones and Bartlett Publishers, Boston, MA, 1992.
*A finitely generated nilpotent group has a Dehn function equivalent to ''n''^''d'' where ''d'' ≥ 1 and all positive integers ''d'' are realized in this way. Moreover, every finitely generated nilpotent group ''G'' admits a polynomial isoperimetric inequality of degree ''c'' + 1, where ''c'' is the nilpotency class of ''G''.
*The set of real numbers ''d'' ≥ 1, such that there exists a finitely presented group with Dehn function equivalent to ''n''^''d'', is dense in the interval asymptotic_cones_of_a_finitely_presented_group_are_simply_connected_space.html" "title="ultralimit.html" ;"title=",\infty).
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