A primer to geometric group theory

Yann Ollivier

Contents

Free groups, group presentations. Cayley graphs and complexes.Groups as geometric objects. Van Kampen diagrams. Hyperbolicgroups. Suggested reading.

This text, originally written as an introduction to my book on random groups,gives an introduction to the study of discrete groups from a geometric point of view.It contains basic facts on group presentations and free groups; the construction ofCayley graphs and complexes; a sketch of what van Kampen diagrams are; andthe different definitions of hyperbolic groups. An explicit example then gathers itall in two pictures.

We do not pretend to give an overview of geometric group theory as a whole,and many important aspects of the field are not mentioned.

A few selected references are given at the end as suggested reading.

Free groups, group presentations

From a geometric group theorist’s viewpoint, which may not be everyone’s, thesimplest of all groups are free groups over some set of generators.

Let S+ be any set (frequently S+ = {a1, . . . , am} is finite). Intuitively speaking,the free group on S+ is the group consisting of all formal products of elements ofS+ and their formal inverses, with the cancellation xx−1 = x−1x = e as the onlycomputation rule.

More precisely, let S− be the set of formal inverses x−1 of the elements x ∈ S,which is just a distinct copy of S+, and let S = S+ ⊔ S−. For n ∈ N, a word oflength n on the alphabet S is a sequence of n elements of S. In the particular casen = 0 there is only one such sequence, called the empty word. A word on S is justa word of any length, i.e. an element of

⊔

n=0,1,2... Sn.

A word is said to be reduced if it does not contain as a subword any sequence ofthe form xx−1 or x−1x, for any x ∈ S. With any word can be associated a reducedword, by iterated removal of all such cancellable pairs (the reduced word obtained

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does not depend on the order in which removals are performed), an operationcalled reduction.

The free group generated by S (or by S+, terminology is floppy) has all reducedwords on S as its elements. Multiplication is simply the concatenation of words,followed by reduction if necessary. The neutral element e is the empty word.

Of special importance is the case when S = {a1, . . . , am} is finite; the corre-sponding free group is denoted by Fm. When m = 1 it is isomorphic to the groupof integers Z.

Now any group can be seen as a free group but with more “computation rules”than simply xx−1 = x−1x = e. This gives rise to the notion of group presentations:a group specified by a given set of generators S, with some“enforced” computationrules. For example, the presentation 〈 a | a3 = e 〉 (read: the group generated bya, knowing that a3 = e) defines a group isomorphic to Z/3Z.

Namely, a computation rule is any equality w1 = w2 where w1, w2 are wordson a given alphabet S. Any such rule can be rewritten w1w

−1

2 = e, and so most ofthe time rules are specified by giving only one word r, with the rule r = e in mind.

So let R be a set of words on a given alphabet S = S+⊔S−. The group presentedby 〈S+ | {r = e}r∈R 〉, or, more simply, by 〈S+ | R 〉, is defined as follows. Startfrom the free group FS generated by S+. The way to enforce a relation r = e isto quotient FS by the normal subgroup generated by r. So let 〈R〉 be the normalclosure of the subgroup of FS generated by all words r ∈ R. The group presentedby 〈S+ | R 〉 is the group FS/〈R〉. It is the “largest” group in which all relationsr = e, r ∈ R, hold.

Let us give a few examples: the free group F1 of rank one (a.k.a. Z) has thepresentation 〈 a | ∅ 〉. The cyclic groups Z/nZ are given by 〈 a | an = e 〉. Thefree group of rank two is F2 = 〈 a, b | ∅ 〉; if we force a and b to commute we getZ × Z = 〈 a, b | ab = ba 〉.

The elements of S+ in a presentation 〈S+ | R 〉 are called generators. Those ofR are called relators. It is easily seen that relators can always be assumed to bereduced words.

Note that any group has some presentation, in a kind of tautological way. LetG be a group and take S+ = G i.e. all elements of G will be generators. Now let theset of words R consist of all products of elements of S which happen to be equalto e in G. It is easy to check that G = 〈S+ | R 〉. (Of course this presentation is,in general, way too large.)

This means that free groups have a“universal”property, namely, for each groupG there is a set S and a surjective homomorphism from a free group FS to G. Moreprecisely, if X ⊂ G is any set which generates G together with X−1, then there is asurjective homomorphism FX → G“sending X to X” i.e. sending an abstract word

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in the generators to its image in G. If R is the kernel of this homomorphism, thenany part R ⊂ R the normal closure of which is R, will give rise to a presentationG = 〈X | R 〉.

A group is finitely presented if it admits a presentation 〈S+ | R 〉 with both S+

and R finite. A group is finitely generated (or of finite type) if S+ is finite.Presentations of a given group are by no means unique. For example, the trivial

group {e} has arbitrarily (not so) stupid presentations such as 〈 a, b | a = e, b2ab−1 = e 〉.(In fact it is not even algorithmically decidable whether a given presentation definesthe trivial group or not!)

Cayley graphs and complexes

Hereafter, in order to get locally compact objects, we suppose that all sets ofgenerators S are finite.

Cayley graphs. The above may seem quite combinatorial but actually carriesa geometrical meaning, which is sometimes a more natural way to think of grouppresentations. For example, the group given by 〈 a | a3 = e 〉 can be thought of asa three-edge cycle.

Let G be a group generated by a set S = S+ ⊔ S−. The Cayley graph of Gwith respect to S is the graph with elements of G as vertices and in which edgescorrespond to multiplication on the right by the generators.

More precisely, the Cayley graph is an unoriented graph with some decoration.Vertices are the elements of G. Now for each x ∈ G and s ∈ S+, add an unorientededge between the vertices x and xs, so that edges are in bijection with G×S+ (thismay result in multiple edges and loops). Now keep track of this on the edges, bydeciding that with each edge together with an orientation choice, will be associateda label in S+ ⊔ S−; namely, the edge from x to xs, oriented this way, will havelabel s, and label s−1 when oriented the other way around1.

Basic examples are (w.r.t. the obvious generating sets): The Cayley graph ofthe free group Fm is an infinite tree in which each vertex has valency 2m. TheCayley graph of the group Z×Z is an infinite square grid in the plane. The Cayleygraph of the cyclic group of order n is an n-edge cycle.

The Cayley graph is homogeneous: all vertices play the same role and couldhave been chosen to represent the neutral element e.

Given any vertex x in the Cayley graph and any word w on the alphabet S,we can start at x and track w in the graph by “following the edge labels”, which

1The details of these definitions guarantee that each vertex has valency 2×#S+ and that theCayley graph of the cyclic group of order n is an n-cycle even for n = 1, 2.

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brings us at xw. This path is called the lift of w (starting at x). Note that w isreduced if and only if this path has no local backtracks.

Left and right actions. The group G acts on the vertices of the Cayley graphin two ways: by left or right multiplication.

Left multiplication by g ∈ G is a graph action: it brings adjacent vertices toadjacent vertices, preserving edges. In particular it preserves the graph distance.Note however that it does not bring a point to a nearby point: x and gx can bevery far away in the graph.

Conversely, right multiplication by g ∈ G is not, in general, a graph action,because a priori one does not pass from xg to xsg by multiplication by s. So twopoints linked by an edge are not mapped to two points linked by an edge: rightmultiplication acts only at the level of vertices of the Cayley graph. However, rightmultiplication by g moves a given point by a distance at most the length of g (sinceit corresponds to following the path labelled by g in the Cayley graph, starting atthe given point).

Consequently, when one mentions the action of G on its Cayley graph, it is theleft action which is meant. (Test all this in F2...)

Cayley complexes. It is worth noting that any word equal to e in G will liftto a loop in the Cayley graph, and conversely. If G is given by a presentation〈S+ | R 〉, this will be the case, in particular, for the relators r ∈ R.

The Cayley complex of a presentation G = 〈S+ | R 〉 is a 2-dimensional complexobtained by gluing a disk on all paths of the Cayley graph labelled by a relatorr ∈ R. More precisely, to each x ∈ G and r ∈ R, consider the lift of r startingat x in the Cayley graph, which is a closed path, and glue a disk along this path.Consequently the set of faces of the Cayley complex is in bijection with G × R.

Basic examples are (w.r.t. the usual presentations): The Cayley complex of afree group is just its Cayley graph. The Cayley complex of Z × Z is the squaretiling of the plane. The Cayley complex of the cyclic group of order n consists ofn disks sharing a common boundary (a copy of the relation an = e is glued to eachelement)2.

Most importantly, the Cayley complex is simply connected. Indeed, loops inthe Cayley graph label words equal to the identity. Since by definition, relatorsgenerate all relations in the group, such words are exactly products of (conjugates

2Several authors, including Lyndon–Schupp (1997), take another convention when some rela-tor is a proper power, in order to avoid this multiplicity. This sometimes leads to contradictionsin their exposition, such as in Prop. III.4.3 of Lyndon–Schupp. With the convention used here,the Cayley complex is the universal covering of the 2-skeleton of the standard classifying spaceassociated with the presentation.

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of) relators in the presentation. We precisely added disks along loops of the Cayleygraph representing the relators, making these loops homotopically trivial.

A word about classifying spaces. Here is another method to define the Cayleygraph and complex. Namely, consider a group presentation G = 〈S+ | R 〉. Thereis a standard way to get a topological space with fundamental group G. Start witha bouquet B of #S+ circles, that is, a graph made of one single vertex (denotede) and #S+ unoriented loops. Choose an orientation for each loop and label loopsbijectively by the generators in S+; consider that the loop with reverse orientationbears the inverse label in S−.

The fundamental group of this bouquet of circles is the free group with #S+

generators. The universal cover of this labelled graph is exactly the Cayley graphof the free group on S+.

Now add “relations” in the following way. Any word on the alphabet S+ ⊔ S−

lifts to a closed path in the labelled graph B, just as above for Cayley graphs.For each relator r ∈ R, add a disk to B along the path labelled by r. In thefundamental group of B, this has the effect of killing the element r. Thus at theend, one gets a 2-complex B (with one vertex, #S+ edges and #R faces) thefundamental group of which is precisely the group G = 〈S+ | R 〉. The universalcover of B is exactly the Cayley complex of this presentation.

(By the way, if one goes on with this process and kill all the higher-dimensionalhomotopy groups of B by adding sufficiently many balls of dimension > 3, one getsa so-called classifying space for the group G, i.e. a space BG with fundamentalgroup G and such that the universal cover EG is contractible.)

Groups as geometric objects

Here again all groups are assumed to be finitely generated.

Metrics on groups. The Cayley graph of a group w.r.t. some generating set isnaturally a metric space, defining each edge to have length 1.

The combinatorial way to look at this is as follows: Given a group G generatedby a set of elements S+, it is natural to define the norm ‖g‖ of an element g ∈ G asthe smallest length of a word expressing g as a product of generators in S+ ⊔ S−.This coincides, of course, with the graph distance from g to e in the Cayley graphof G w.r.t. this generating set. Note that ‖g‖ = ‖g−1‖, and of course ‖gg′‖ 6

‖g‖ + ‖g′‖.So we get a distance function on G by setting dist(g, h) to be the graph distance

from g to h in the Cayley graph. Since edges of the Cayley graph correspond toright multiplication by a generator, this is the smallest length of a word w such

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that h = gw, that is, dist(g, h) = ‖g−1h‖. The two properties of ‖·‖ mentionedabove are just the usual distance axioms.

As mentioned above, the left action of G on itself preserves this metric.

Changing generators. Of course this distance depends on the chosen generat-ing set. A finitely generated group is thus not equipped with a canonical metric,but with a family of metrics associated with all possible finite generating sets.These metrics are related in some way, which we explore now.

Let S+ = {s1, . . . , sn} and S ′+ = {s′1, . . . , s

′p} be two finite generating sets for a

group G; let ‖·‖ and ‖·‖′ be the two associated norms. We want to control ‖·‖′ interms of ‖·‖.

Since S+ is a generating set, each of the elements in S ′+ has a writing in terms

of the generators in S+ ⊔ S−. Let K be the largest length of such a writing i.e.K = maxs′∈S′

+‖s′‖. Since S ′

+ is finite, we have K < ∞.Now suppose that some element g has a writing of length n in terms of S ′

+⊔S ′−.

By replacing each element of S ′+ by a writing of it in terms of S+ ⊔ S−, we get

a writing of g of length at most Kn in terms of S+ ⊔ S−. So ‖·‖ 6 K ‖·‖′. Thereasoning is two-sided, and so we get that the metrics defined by S+ and S ′

+ arebi-Lipschitz equivalent.

Quasi-isometries. Actually a slightly looser definition of equivalence betweenmetric spaces than bi-Lipschitz equivalence has proven very fruitful: that of quasi-isometry. It allows, for example, the spaces R and Z to be quasi-isometric, byneglecting what happens at small scales.

Let (X, dX) and (Y, dY ) be two metric spaces. They are quasi-isometric if thereexist two maps f : X → Y and g : Y → X which distort distances in a linearlycontrolled way and which are almost inverse to each other (up to a finite distanceerror). That is, there exist constants λ > 0, C > 0 such that

dY (f(x), f(x′)) 6 λ dX(x, x′) + C

dX(g(y), g(y′)) 6 λ dY (y, y′) + C

dX(g(f(x)), x) 6 C

dY (f(g(y)), y) 6 C

for all x, x′ ∈ X, y, y′ ∈ Y . This is an equivalence relation.This notion is relevant for unbounded spaces only: any bounded metric space

is quasi-isometric to a point.A change in the generating set of a group is in particular a quasi-isometry. So

any quasi-isometry invariant of a metric space, when applied to Cayley graphs,will provide a well-defined invariant of finitely generated groups.

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Van Kampen diagrams

Van Kampen diagrams are a visual way to represent how all equalities holding ina group are derived from combinations of relators.

Let G = 〈S+ | R 〉 be a group presentation. Since we only have access toelements of G as products of generators, we want to know when two words representthe same element of G, i.e. we are interested in the set of equalities of words x = ythat hold in G. Since this can be rewritten as xy−1 = e, it is enough to determinethe set of words representing the identity element of G (so-called word problem).

This problem is actually algorithmically unsolvable: there is no algorithm that,given any finite presentation and any word, always answers whether the given wordis equal to e in the presentation. Moreover, this already holds for some fixed groups:there exist some group presentations for which there is no algorithm which, givenany word, answers whether or not it is equal to e in the presentation.

A word is equal to e in the group G presented by 〈S+ | R 〉 = FS+/〈R〉, by

definition, if it lies in the kernel of the map FS+→ G, that is, in the normal

subgroup 〈R〉 generated by the relators. Hence, a word w is equal to e in G if andonly if, as a word, it can be written as a product of conjugates of relators:

w =N∏

i=1

ui r±1

i u−1

i

where ri ∈ R and ui is any word. (The number N of relators in this decomposition,which depends on w, will play an important role below in the definition of hyper-bolic groups.) So algebraically speaking, the set of consequences of the relators isthe normal closure of the subgroup they generate. Van Kampen diagrams are avisual interpretation of these products of conjugates.

Giving a topologically clean definition of van Kampen diagrams is beyond thescope of this review. Basically, the idea is to consider each relator r ∈ R as apolygon with as many edges as letters in r, the edges of which are labelled withthe successive letters of r (the inverse of r may also be used to build a polygonwith reverse orientation). Here is the polygon associated with the simple relatoraba−1b−1 (starting at bottom left corner, counterclockwise):

b

a

a

b

Now polygons bearing (same or different) relators can be glued to each otheralong edges bearing the same letter. Van Kampen diagrams are the figures re-

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sulting from such connected, simply connected gluings of relator-bearing poly-gons. For example, the following van Kampen diagram w.r.t. the presentation〈 a, b | ab = ba 〉 (i.e. with the only relator aba−1b−1) is a visual proof that if acommutes with b, then so does a2.

b

a

a

a

a

bb

An important theorem of van Kampen states that the boundary words of suchplanar diagrams are exactly those words equal to e in the presentation.

The connection with products of conjugates of relators is clear on the fol-lowing picture, which builds the boundary word b−1a2ba−2 of the van Kampendiagram given above (starting at top left corner, counterclockwise) as a productb−1rb (ab−1)r(ab−1)−1 of conjugates of the relator r = aba−1b−1. Shrinking thisdiagram (by identifying edges arising from the same point with the same label)produces the two-square one above.

bb

a

a a

aa

bb

b b

Incidentally, coming back to the algorithmic decidability problems mentionedabove, we see that the word problem is semi-decidable: if a word is indeed equalto e, then, searching through all van Kampen diagrams, we will eventually findone decomposing the word as a product of conjugates of relators; but proving thatthe word is not equal to e would a priori require examination and rejection of allpossible diagrams.

Hyperbolic groups

A class of groups the interest for which has never declined over the years sinceits introduction by Gromov is that of hyperbolic groups. They have a combina-torial definition, word hyperbolicity, and a geometric one, δ-hyperbolicity. Since(spoiler!) these two notions have turned out to be equivalent, we simply use theterm hyperbolic.

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Word hyperbolicity. Let G = 〈 a1, . . . , am | R 〉 be a group with finite presen-tation. Recall that any word w equal to e in G can be written as a product

w =N∏

i=1

uir±1

i u−1

i

(generally not in a unique way). Let N(w) be the minimal number of relators insuch a writing of w. A very natural question is: How does N(w) behave when thelength of w grows larger and larger?

The presentation is said to be word-hyperbolic if N(w) grows at most linearlywith the length of w, that is, if there exists a constant C such that for any w wehave N(w) 6 C |w|.

It is not difficult to see that for finite presentations, this linearity does notdepend on the presentation chosen for a given group G: indeed, each generator inthe second presentation is a product of (finitely many) generators in the first one,and each relator in the second presentation is a consequence of (can be written asproducts of conjugates of) finitely many relators in the first one, so that, since thenumber of generators and relators is finite, the constant C is perturbed by at mostthese quantities. So this yields a well-defined notion of a word-hyperbolic group.

When thought of in terms of van Kampen diagrams, this reads as follows: If wis a word equal to e in the group, then there is a van Kampen diagram D with was its boundary word. Now by definition N(w) is the number of faces |D| of thisvan Kampen diagram and the length of w is the boundary length |∂D|. So theword hyperbolicity condition rewrites as

|D| 6 C |∂D|

which is a linear isoperimetric inequality for van Kampen diagrams. Linear isoperi-metric inequalities are a negative curvature phenomenon: for example, they aresatisfied by domains in the hyperbolic plane (with area standing for |D| and bound-ary length standing for |∂D|), but not by domains in the Euclidean plane (whereboundary length grows only like the square root of area).

δ-hyperbolicity. This is a more general notion defined in any geodesic space(that is, a metric space in which the distance between two points is realized byone or several paths between them, called geodesic segments). Since any graph isa geodesic space (by definition, edge-paths realize the distance), it can be appliedto Cayley graphs of groups.

δ-hyperbolicity is a way to measure how “negatively curved” or “tree-like” aspace looks at large scale. In a traditional negative curvature setting, triangleshave the property that the sum of their angles is less than 2π: they are curvedinwards. This is measured by Rips’ condition of thinness of triangles.

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A triangle in a geodesic space is specified by a triple of points (x, y, z) togetherwith three geodesic segments joining them pairwise, noted [xy], [yz] and [zx]. Forδ > 0, the triangle is said to be δ-thin if any point on [xy] lies at distance at mostδ from the union [yz]∪ [zx] of the two other sides, and similarly for points on [yz],on [zx]. That is, the “gap” at the middle of the triangle has size roughly δ: eachside does not depart too much from the other two.

δδ

For δ > 0, a geodesic space is said to be δ-hyperbolic (or simply hyperbolic ifδ is not specified) if any triangle in it is δ-thin. Note that this must hold for allchoices of geodesics between the vertices in case there are several.

The usual hyperbolic plane is hyperbolic indeed (δ = 10 works). Any boundedmetric space is δ-hyperbolic (take its diameter as δ). The Euclidean plane or thegrid Z × Z are not hyperbolic for any δ > 0.

Another important example is a (finite or infinite) tree, which is 0-hyperbolic(triangles are flattened). In fact, δ-hyperbolic spaces are those which, “seen fromfar away”, look like trees; they can actually by approximated by trees (up to δ) ina very precise sense.

Now a finite group presentation is δ-hyperbolic if the associated Cayley graph is.The most basic examples (apart from finite groups) are the free groups Fm, whoseCayley graphs w.r.t. the standard generating set are trees, hence 0-hyperbolic.Hyperbolic groups are thus a natural geometric generalization of free groups.

Importantly (but not obviously), δ-hyperbolicity for some δ > 0 is preservedunder quasi-isometries, although the value of δ may change. In particular, for agroup it does not depend on the choice of a generating set.

Hyperbolicity. A finitely presented group is word-hyperbolic if and only if it isδ-hyperbolic for some δ > 0. This is a non-trivial theorem and may actually bethe shortest way to show that δ-hyperbolicity does not depend on the presentationof a given group (though the value of δ does).

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Note that we have defined word hyperbolicity only for group presentations,whereas δ-hyperbolicity makes sense in a more general context; but the notion oflinear isoperimetric inequality of van Kampen diagrams can be extended to anygeodesic space and is still equivalent to δ-hyperbolicity.

Small cancellation. The small cancellation conditions are simple combinatorialcriteria on a group presentation which imply hyperbolicity. Maybe the one mostfrequently encountered is the C ′(α) condition. Remember the idea of van Kampendiagrams: relators in a presentation are represented as polygons with boundarylabelled by the relator. Now if two such polygons have a common subword on theirboundary (more precisely, if there is a word w such that w is a subword of theboundary word of the first polygon and w−1 is a subword of the boundary wordof the second one), we can glue them along this subword to form a two-face vanKampen diagram.

The C ′(α) condition for a presentation (for 0 < α < 1) demands that any suchgluing between two polygons bearing relators r1 and r2 in the presentation occursalong a path w of length less than α times the infimum of the lengths of r1 andr2 (except for the trivial “degenerate” gluing between a polygon and the same onewith inverse orientation i.e. r2 = r−1

1 ).If a group presentation satisfies the C ′(1/6) condition, then the group is hyper-

bolic. This results from a simple Euler characteristic argument on van Kampendiagrams, which allows to show that those have enough boundary edges to ensurea linear isoperimetric inequality.

The limit case on which the significance of 1/6 is clear is the standard hexagonaltiling of the plane, which satisfies C ′(1/6 + ε) for any ε > 0 but not C ′(1/6), andwhich is not hyperbolic (compare the heptagonal tiling of the hyperbolic plane).

All in one: an example. Let G be the group presented by⟨

a, b, c, d | aba−1b−1cdc−1d−1 = e⟩

which means that the only polygon appearing in van Kampen diagrams is thefollowing octogon.

c

c

d

d

a

b

a

b

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Now the only ways to glue such a polygon with a copy of itself consist in gluingthe two a’s, or gluing the two b’s, or the two c’s, or the two d’s, but there is noway to find a gluing along two consecutive letters. Since the length of the relatoris 8, this means that this presentation satisfies the C ′(1/8 + ε) small cancellationcondition (but not C ′(1/8)). Since 1/8 < 1/6 the criterion above applies and sothis group is hyperbolic.

This example is not anecdotic. Copies of the above octogon can be glued (inthe allowed ways: an a with an a, etc.) to form the standard octogonal tilingof the hyperbolic plane. Actually this tiling is exactly the Cayley graph of thegroup, which we represent on pages 13 and 14 in two different Euclidean views:a Poincare model centered either on a face, emphasizing the tiling aspect, or onan element, emphasizing the Cayley graph. (When looking at the figures, keep inmind that the hyperbolic metric goes to infinity close to the disk boundary, so thatall octogons are actually isometric and play the same role. The second picture isobtained from the first one by performing a hyperbolic isometry sending the leftvertex of the large bottom a to the center of the disk.)

Since the hyperbolic plane is hyperbolic, this is another way to check hyperbol-icity of this group: the group identifies with vertices of the tiling, equipped withthe edge metric, which is a discrete subset quasi-isometric to the whole hyperbolicplane. Note how tree-like the Cayley graph looks in spite of the presence of cyclesof length 8.

A group naturally acts on its Cayley graph by left multiplication and so we cantake the quotient of this hyperbolic tiling by action of the group. This provides atwo-dimensional object, the fundamental group of which is the group.

Let us take a closer look at this quotient. A single octogon is a fundamentaldomain for the action of the group on the tiling (i.e. the set of all translates of someoctogon by the group exactly produces the tiling) and so the quotient is obtainedby taking a single octogon and identifying its edges according to the labels. So takea single copy of the octogon above and twist it in three-dimensional space so thatthe two edges labelled by a are identified (preserving the orientation implied bythe arrows); then identify the two edges labelled by b, then the two ones labelledby c, then the ones labelled by d (all eight vertices will be identified along theway). This actually leaves us with a surface of genus two, i.e. the gluing of twotori (this is quite hard to figure out—check it first in the simpler case of a squareaba−1b−1 instead of the octogon above: this square transforms into a torus).

So the group can be represented as the fundamental group of a surface of genus2. This surface inherits a metric of negative curvature from that of the hyperbolicplane. Actually, fundamental groups of negatively curved compact surfaces, or ofpolyhedra with negative curvature in some combinatorial sense, were an importantmotivation for the introduction of hyperbolic groups and initially the main sourceof examples.

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14

Suggested reading

Groups as geometric objects, hyperbolic groups, quasi-isometries, Cayley graphs...E. Ghys, P. de la Harpe, Sur les groupes hyperboliques d’apres Mikhael Gromov,Progress in Math. 83, Birkhauser (1990).

Groups as geometric objects, quasi-isometries...M. Gromov, Infinite groups as geometric objects, in Proceedings of the InternationalCongress of Mathematicians, Vol. 1, 2, Warsaw (1983), 385–392, PWN, Warsaw,1984.

Group presentations, free groups, van Kampen diagrams (under the name “cancel-lation diagrams”), decision problems...J. J. Rotman, An introduction to the theory of groups, fourth edition, GraduateTexts in Mathematics 148, Springer (1995), especially chapters 11 and 12.

Group presentations, free groups, van Kampen diagrams, small cancellation the-ory...R. C. Lyndon, P. E. Schupp, Combinatorial group theory, Ergebnisse der Mathe-matik und ihrer Grenzgebiete 89, Springer (1977).

Negative curvature, hyperbolic groups, isoperimetric inequalities...M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehrender mathematischen Wissenschaften 319, Springer (1999), especially chapters Hand Γ.

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