Dehn function#Known results

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.Martin Greendlinger, [https://archive.today/20130105075745/http://www3.interscience.wiley.com/journal/113397463/abstract?CRETRY=1&SRETRY=0 Dehn's algorithm for the word problem.] Communications on Pure and Applied Mathematics, vol. 13 (1960), pp. 67–83. 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. {{ISBN|0-387-96618-8}}.{{MR|0919829}} 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 showedM. Sapir, J.-C. Birget, E. Rips. Isoperimetric and isodiametric functions of groups. Annals of Mathematics (2), vol 156 (2002), no. 2, pp. 345–466. that most "reasonable" time complexity functions of Turing machines can be realized, up to natural equivalence, as Dehn functions of finitely presented groups.

Let

:$G=\backslash langle\; X|R\backslash rangle\backslash qquad\; (*)$

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.

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

:$w=u\_1r\_1u\_1^\{-1\}\backslash cdots\; u\_m\; r\_mu\_\{m\}^\{-1\}\; \backslash text\{\; in\; \}\; F(X),$   (♠)

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.

An isoperimetric function for a finite presentation (∗) is a monotone non-decreasing function

:$f:\; \backslash mathbb\; N\backslash to\; [0,\backslash infty)$

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.

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

:$\{\backslash rm\; Dehn\}(n)=\backslash max\backslash \{\{\backslash rm\; Area\}(w):\; w=1\; \backslash text\{\; in\; \}\; G,\; |w|\backslash le\; n,\; w\; \backslash text\{\; freely\; reduced\}.\backslash \}$

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.

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

:$f,g:\; \backslash mathbb\; N\backslash to\; [0,\backslash infty)$

one says that f is dominated by g if there exists C ≥1 such that

:$f(n)\backslash le\; Cg(Cn+C)+Cn+C$

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 aⁿ ≈ bⁿ. 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.

• 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, [https://books.google.com/books?id=kLTKEymaDO0C&dq=Higher-dimensional+isoperimetric+(or+Dehn)+functions+of+groups&pg=PA79 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.Juan M. Alonso, Inégalités isopérimétriques et quasi-isométries. Comptes Rendus de l'Académie des Sciences, Série I, vol. 311 (1990), no. 12, pp. 761–764. 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.

• For any finite presentation of a finite group G we have Dehn(n) ≈ n.Martin R. Bridson. [https://books.google.com/books?id=VB_f6xnmvikC&dq=The+geometry+of+the+word+problem.+Invitations+to+geometry+and+topology,&pg=PA29 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. {{ISBN|0-19-850772-0}}.
• For the closed oriented surface of genus 2, the standard presentation of its fundamental group

:$G=\backslash langle\; a\_1,a\_2,b\_1,b\_2|[a\_1,b\_1][a\_2,b\_2]=1\backslash rangle$

:satisfies Dehn(n) ≤ n and Dehn(n) ≈ n.

• For every integer k ≥ 2 the free abelian group $\backslash mathbb\; Z^k$ has Dehn(n) ≈ n².
• The Baumslag-Solitar group

:$B(1,2)=\backslash langle\; a,b|\; b^\{-1\}ab=a^2\backslash rangle$

:has Dehn(n) ≈ 2ⁿ (see S. M. Gersten, Dehn functions and l₁-norms of finite presentations. Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), pp. 195–224,

Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992. {{ISBN|0-387-97685-X}}.).

• The 3-dimensional discrete Heisenberg group

:$H\_3=\backslash langle\; a,b,\; t|\; [a,t]=[b,t]=1,\; [a,b]=t^2\; \backslash rangle$

:satisfies a cubic but no quadratic isoperimetric inequality.

• Higher-dimensional Heisenberg groups

:$H\_\{2k+1\}=\backslash langle\; a\_1,b\_1,\backslash dots,\; a\_k,b\_k,t|\; [a\_i,b\_i]=t,\; [a\_i,t]=[b\_i,t]=1,\; i=1,\backslash dots,\; k,\; [a\_i,b\_j]=1,\; i\backslash ne\; j\backslash rangle$,

:where k ≥ 2, satisfy quadratic isoperimetric inequalities.D. Allcock, [https://doi.org/10.1007%2Fs000390050053 An isoperimetric inequality for the Heisenberg groups.] Geometric and Functional Analysis, vol. 8 (1998), no. 2, pp. 219–233.

• 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 n² (see V. S. Guba, [https://doi.org/10.1007%2Fs00222-005-0462-z The Dehn function of Richard Thompson's group F is quadratic.] Inventiones Mathematicae, vol. 163 (2006), no. 2, pp. 313–342.).
• The so-called Baumslag-Gersten group

::$G=\backslash langle\; a,\; t|\; (t^\{-1\}a^\{-1\}\; t)\; a\; (t^\{-1\}\; at)=a^2\backslash rangle$

: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 A. N. Platonov, An isoparametric function of the Baumslag-Gersten group. (in Russian.) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2004, no. 3, pp. 12–17; translation in: Moscow University Mathematics Bulletin, vol. 59 (2004), no. 3, pp. 12–17 (2005).).

• 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.A. Yu. Olʹshanskii. Hyperbolicity of groups with subquadratic isoperimetric inequality. International Journal of Algebra and Computation, vol. 1 (1991), no. 3, pp. 281–289. {{MR|1148230}}{{doi|10.1142/S0218196791000183}}B. H. Bowditch. [http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.mmj/1029005156&page=record A short proof that a subquadratic isoperimetric inequality implies a linear one.]

Michigan Mathematical Journal, vol. 42 (1995), no. 1, pp. 103–107. {{MR|1322192}}{{doi|10.1307/mmj/1029005156}} 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. {{ISBN|0-86720-244-0}} {{MR|1161694}}
• 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.S. M. Gersten, D. F. Holt, T. R. Riley, [https://doi.org/10.1007%2Fs00039-003-0430-y Isoperimetric inequalities for nilpotent groups.] Geometric and Functional Analysis, vol. 13 (2003), no. 4, pp. 795–814. {{MR|2006557}}{{doi|10.1007/s00039-003-0430-y}}
• 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 $[2,\backslash infty)$.N. Brady and M. R. Bridson,

[https://doi.org/10.1007%2FPL00001646 There is only one gap in the isoperimetric spectrum.]

Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1053–1070.

• If all asymptotic cones of a finitely presented group are simply connected, then the group satisfies a polynomial isoperimetric inequality.M. Gromov, Asymptotic invariants of infinite groups, in: "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295.
• If a finitely presented group satisfies a quadratic isoperimetric inequality, then all asymptotic cones of this group are simply connected.P. Papasoglu. [http://intlpress.com/journals/JDG/archive/pdf/1996/44-4-789.pdf On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality.] {{Webarchive|url=https://web.archive.org/web/20110523202024/http://intlpress.com/journals/JDG/archive/pdf/1996/44-4-789.pdf |date=2011-05-23 }} Journal of Differential Geometry, vol. 44 (1996), no. 4, pp. 789–806.
• If (M,g) is a closed Riemannian manifold and G = π₁(M) then the Dehn function of G is equivalent to the filling area function of the manifold.J. Burillo and J. Taback. [http://nyjm.albany.edu/j/2002/8-11.pdf Equivalence of geometric and combinatorial Dehn functions.] New York Journal of Mathematics, vol. 8 (2002), pp. 169–179.
• If G is a group acting properly discontinuously and cocompactly by isometries on a CAT(0) space, then G satisfies a quadratic isoperimetric inequality.M. R. Bridson and A. Haefliger, [https://books.google.com/books?id=3DjaqB08AwAC&q=.+R.+Bridson+and+A.+Haefliger,+%22Metric+spaces+of+non-positive+curvature%22 Metric spaces of non-positive curvature.] Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer-Verlag, Berlin, 1999. {{ISBN|3-540-64324-9}}; Remark 1.7, p. 444. In particular, this applies to the case where G is the fundamental group of a closed Riemannian manifold of non-positive sectional curvature (not necessarily constant).
• The Dehn function of SL(m, Z) is at most exponential for any m ≥ 3.{{cite journal | url=http://www.sciencedirect.com/science/article/pii/S0926224504000087 | doi=10.1016/j.difgeo.2004.03.001 | title=On polyhedral retracts and compactifications of locally symmetric spaces | journal=Differential Geometry and Its Applications | date=May 2004 | volume=20 | issue=3 | pages=293–318 | last1=Leuzinger | first1=Enrico }} For SL(3,Z) this bound is sharp and it is known in that case that the Dehn function does not admit a subexponential upper bound. The Dehn functions for SL(m,Z), where m > 4 are quadratic.Robert Young, [http://annals.math.princeton.edu/2013/177-3/p04 The Dehn function of SL(n;Z).] Annals of Mathematics (2), vol. 177 (2013) no.3, pp. 969–1027. The Dehn function of SL(4,Z), has been conjectured to be quadratic, by Thurston. This and, more generally, Gromov's conjecture that lattices in higher rank Lie groups have a quadratic Dehn function has been proved by Leuzinger and Young.E. Leuzinger and R. Young, [https://doi.org/10.4007/annals.2021.193.3.2 Filling functions of arithmetic groups.] Annals of Mathematics, vol. 193 (2021), pp. 733–792.
• Mapping class groups of surfaces of finite type are automatic and satisfy quadratic isoperimetric inequalities.Lee Mosher, [https://www.jstor.org/stable/2118637 Mapping class groups are automatic.] Annals of Mathematics (2), vol. 142 (1995), no. 2, pp. 303–384.
• The Dehn functions for the groups Aut(F_k) and Out(F_k) are exponential for every k ≥ 3. Exponential isoperimetric inequalities for Aut(F_k) and Out(F_k) when k ≥ 3 were found by Hatcher and Vogtmann.Allen Hatcher and Karen Vogtmann,

[http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102365632 Isoperimetric inequalities for automorphism groups of free groups.] Pacific Journal of Mathematics, vol. 173 (1996), no. 2, 425–441. These bounds are sharp, and the groups Aut(F_k) and Out(F_k) do not satisfy subexponential isoperimetric inequalities, as shown for k = 3 by Bridson and Vogtmann,Martin R. Bridson and Karen Vogtmann, [https://archive.today/20120713005504/http://blms.oxfordjournals.org/cgi/content/abstract/27/6/544 On the geometry of the automorphism group of a free group.] Bulletin of the London Mathematical Society, vol. 27 (1995), no. 6, pp. 544–552. and for k ≥ 4 by Handel and Mosher.Michael Handel and Lee Mosher, [https://msp.org/gt/2013/17-3/p06.xhtml Lipschitz retraction and distortion for subgroups of Out(F_n).] Geometry & Topology, vol. 17 (2013), no. 3, pp. 1535–1579. {{MR|3073930}}{{doi|10.2140/gt.2013.17.1535}}

• For every automorphism φ of a finitely generated free group F_k the mapping torus group $F\_k\backslash rtimes\_\backslash phi\; \backslash mathbb\; Z$ of φ satisfies a quadratic isoperimetric inequality.Martin R. Bridson and Daniel Groves. [http://people.maths.ox.ac.uk/~bridson/papers/BG1/ The quadratic isoperimetric inequality for mapping tori of free-group automorphisms]. Memoirs of the American Mathematical Society, vol 203 (2010), no. 955.
• Most "reasonable" computable functions that are ≥n⁴, can be realized, up to equivalence, as Dehn functions of finitely presented groups. In particular, if f(n) ≥ n⁴ is a superadditive function whose binary representation is computable in time $O\backslash left(\backslash sqrt[4]\{f(n)\}\backslash right)$ by a Turing machine then f(n) is equivalent to the Dehn function of a finitely presented group.
• Although one cannot reasonably bound the Dehn function of a group in terms of the complexity of its word problem, Birget, Olʹshanskii, Rips and Sapir obtained the following result,J.-C. Birget, A. Yu. Ol'shanskii, E. Rips, M. Sapir. Isoperimetric functions of groups and computational complexity of the word problem. Annals of Mathematics (2), vol 156 (2002), no. 2, pp. 467–518. providing a far-reaching generalization of Higman's embedding theorem: The word problem of a finitely generated group is decidable in nondeterministic polynomial time if and only if this group can be embedded into a finitely presented group with a polynomial isoperimetric function. Moreover, every group with the word problem solvable in time T(n) can be embedded into a group with isoperimetric function equivalent to n²T(n²)⁴.

• There are several companion notions closely related to the notion of an isoperimetric function. Thus an isodiametric functionS. M. Gersten, The double exponential theorem for isodiametric and isoperimetric functions. International Journal of Algebra and Computation, vol. 1 (1991), no. 3, pp. 321–327. bounds the smallest diameter (with respect to the simplicial metric where every edge has length one) of a van Kampen diagram for a particular relation w in terms of the length of w. A filling length function the smallest filling length of a van Kampen diagram for a particular relation w in terms of the length of w. Here the filling length of a diagram is the minimum, over all combinatorial null-homotopies of the diagram, of the maximal length of intermediate loops bounding intermediate diagrams along such null-homotopies.S. M. Gersten and T. Riley, [https://doi.org/10.1023%2FA%3A1019682203828 Filling length in finitely presentable groups.] Dedicated to John Stallings on the occasion of his 65th birthday. Geometriae Dedicata, vol. 92 (2002), pp. 41–58. The filling length function is closely related to the non-deterministic space complexity of the word problem for finitely presented groups. There are several general inequalities connecting the Dehn function, the optimal isodiametric function and the optimal filling length function, but the precise relationship between them is not yet understood.
• There are also higher-dimensional generalizations of isoperimetric and Dehn functions.J. M. Alonso, X. Wang and S. J. Pride, [https://doi.org/10.1515/jgth.1999.008 Higher-dimensional isoperimetric (or Dehn) functions of groups.] Journal of Group Theory, vol. 2 (1999), no. 1, pp. 81–112. For k ≥ 1 the k-dimensional isoperimetric function of a group bounds the minimal combinatorial volume of (k + 1)-dimensional ball-fillings of k-spheres mapped into a k-connected space on which the group acts properly and cocompactly; the bound is given as a function of the combinatorial volume of the k-sphere. The standard notion of an isoperimetric function corresponds to the case k = 1. Compared to the classical case only little is known about these higher dimensional filling functions. One chief result is that lattices in higher rank semisimple Lie groups are undistorted in dimensions below the rank, i.e. they satisfy the same filling functions as their associated symmetric space.
• In his monograph Asymptotic invariants of infinite groupsM. Gromov, Asymptotic invariants of infinite groups, in: "Geometric Group Theory", Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993, pp. 1–295. Gromov proposed a probabilistic or averaged version of Dehn function and suggested that for many groups averaged Dehn functions should have strictly slower asymptotics than the standard Dehn functions. More precise treatments of the notion of an averaged Dehn function or mean Dehn function were given later by other researchers who also proved that indeed averaged Dehn functions are subasymptotic to standard Dehn functions in a number of cases (such as nilpotent and abelian groups).O. Bogopolskii and E. Ventura. [https://doi.org/10.1515/JGT.2008.035 The mean Dehn functions of abelian groups.]

Journal of Group Theory, vol. 11 (2008), no. 4, pp. 569–586.Robert Young. [https://archive.today/20120912072112/http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1J-4PT0Y41-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=dd941b487e4fbb13204cf66da8174aaa Averaged Dehn functions for nilpotent groups.] Topology, vol. 47 (2008), no. 5, pp. 351–367.E. G. Kukina and V. A. Roman'kov. [https://doi.org/10.1023%2FA%3A1024776205243 Subquadratic Growth of the Averaged Dehn Function for Free Abelian Groups.] Siberian Mathematical Journal, vol. 44 (2003), no. 4, 1573–9260.

• A relative version of the notion of an isoperimetric function plays a central role in Osin' approach to relatively hyperbolic groups.Densi Osin. [https://books.google.com/books?id=ZTqOG4KLNK4C&dq=osin+relatively+hyperbolic&pg=PA1 Relatively Hyperbolic Groups: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems.] Memoirs of the American Mathematical Society, vol. 179 (2006), no. 843. American Mathematical Society. {{ISBN|978-0-8218-3821-1}}.
• Grigorchuk and Ivanov explored several natural generalizations of Dehn function for group presentations on finitely many generators but with infinitely many defining relations.R. I. Grigorchuk and S. V. Ivanov, [https://doi.org/10.1007%2Fs00039-009-0712-0 On Dehn Functions of Infinite Presentations of Groups], Geometric and Functional Analysis, vol. 18 (2009), no. 6, pp. 1841–1874

• van Kampen diagram
• Word-hyperbolic group
• Automatic group
• Small cancellation theory
• Geometric group theory

{{reflist|2}}

• Noel Brady, Tim Riley and Hamish Short. [https://archive.today/20130202232836/http://www.springerlink.com/content/p8j80k/?sortorder=asc&v=expanded The Geometry of the Word Problem for Finitely Generated Groups.] Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, Basel, 2007. {{ISBN|3-7643-7949-9}}.
• Martin R. Bridson. [https://books.google.com/books?id=VB_f6xnmvikC&dq=The+geometry+of+the+word+problem.+Invitations+to+geometry+and+topology,&pg=PA29 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. {{ISBN|0-19-850772-0}}.

• [https://web.archive.org/web/20090424050849/http://www.maths.bris.ac.uk/~matrr/SL(n,Z).html#14 The Isoperimetric Inequality for SL(n,Z).] A September 2008 Workshop at the American Institute of Mathematics.
• [https://people.maths.ox.ac.uk/bridson/papers/bfs/bfs.pdf PDF of Bridson's article The geometry of the word problem.]

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