This introductory post is based on parts of Martin Bridson’s 2006 ICM article.
Presentations, Cayley graphs, and word metrics
To a topologist’s eye, a finite presentation
for a group is an encoding of a finite 2-dimensional complex known as the presentation 2-complex — a complex with a single vertex, a directed edge for each , and a face for each — a reading of giving that face’s attaching map.
For example, as shown below, is a figure-eight when is , is a torus when is , and is the genus-2 surface when is the surface group .
By the Seifert-van Kampen Theorem, . So appears in a geometric guise as the deck transformations of the universal cover , known asthe Cayley 2-complex. Here is for the three examples.
The 1-skeleton of is the Cayley graph , and the -skeleton can be identified with . The edges of inherit directions and labels from , so we recover what is perhaps the more elementary definition of : the graph with vertex set and, for every and , a directed edge labeled by from to .
The standard path metric on in which each edge has length one, agrees with the word metric on — that for which is the length of the shortest word on the generators and their inverses that represents .
Two key themes of Geometric Group Theory
- Groups act! One tries to extract information about a group from the features of a space on which it acts — information depending heavily on the quality of the action and the richness of the space in question. The space may be a complex such as . It may be the universal cover of a closed Riemannian manifold with fundamental group . It may be a itself, which brings us to —
- Groups are geometric objects. This is already apparent in our discussion — a finitely generated group has a word metric and so is a space… a diffuse cloud of points, perhaps, but when viewed coarsely or on a large-scale, it displays intrinsic and often rich geometry.
Word metrics and Cayley graphs are defined using choices of generating sets. But those choices are downplayed in our large-scale perspective in a manner made precise by the following equivalence relation, quasi–isometry, on metric spaces.
Suppose and . A map between metric spaces is a —quasi-isometric embedding when, for all ,
— that is, the map allows for a bounded amount of stretching and tearing. We say is a —quasi-isometry when, in addition, it is —quasi-onto, i.e. the –neighbourhood of is . Two metric spaces are quasi–isometric when there is some quasi–isometry between them. Pertinent examples of quasi–isometries include:
- For a group with finite generating set , the Cayley graph is quasi–isometric to with the word metric associated to .
- The word metrics associated to any two finite generating sets for a group, yield quasi-isometric metric spaces.
- Finitely generated groups are quasi–isometric to their finite–index subgroups.
Two groups are commensurable when they have finite index subgroups that are isomorphic. So, finitely generated commensurable groups are quasi–isometric.
Bridson imagines mapping the finitely presented groups up to commensurability. The image is reproduced from Martin Bridson’s 2006 ICM article, except an extract from the c. 1430AD “Borgia map” has been inserted in place of a lion in the original, in keeping with the title of this blog.
Key: Ab — abelian, Nilp — nilpotent, PC — polycyclic, Solv — solvable, EA — elementary amenable, F = free, EF — elementarily free, — limit, Hyp — hyperbolic, — CAT(0), SH — semi-hyperbolic, Aut — automatic, IP(2) — quadratic isoperimetric inequality, Comb — combable, Asynch — asynchronously combable, vNT — the von Neumann–Tits line. The question marks indicate regions for which it is unknown whether any groups are present.
The map begins with the finite groups — all are commensurable with the trivial group and so are represented by a single point. Next is , which is surely the most elementary infinite group. From there, Bridson envisages setting out to chart the land in two directions.
Taking a path which favours commutativity, one finds in the corner of the abelian groups, which are a municipality of the nilpotent groups, itself in the town of polycyclic groups, and in the city of solvable groups — extending successively further along what it seems appropriate to call the amenable coastline.
Alternatively, turning away from commutativity, one sets out along the coastline of non–positive curvature. We will survey the classes of groups found there in more detail in a future post, but here’s a brief description. Moving away from we first encounter the finite rank free groups. (This only adds as all those of rank at least 2 are commensurable, but please bear with us). Free groups display tree-like geometry — that is, infinite negative curvature. As such they generalise to hyperbolic groups — negatively curved groups in a strong sense (more on which to follow). Then, relaxing to non-positively curved, one has the CAT(0) groups (that is, groups that act properly cocompactly on CAT(0) spaces). Then the landscape proceeds into regions which represent different ways of drawing out intrinsically group theoretic consequences of the definition of CAT(0) groups — semi-hyperbolic groups, automatic groups, groups enjoying quadratic isoperimetric functions, combable groups, bicombable groups, asynchronously combable groups, etc..
So, what lies midway between these two coastlines? Close to , various classes reach from coast to coast: all the non-positively curved classes include the abelian groups; and there are nilpotent groups of all classes in IP(2) (Young). Sela’s limit groups are another class that deserves to be included — they are an enclave of the CAT(0) groups (Alibegovic and Bestvina) around ; roughly speaking, they are those groups with Cayley graphs that are limits of Cayley graphs of free groups. They include free groups, abelian groups and many surface groups. Remarkably they can be characterised as the finitely generated groups that have the same first order existential theory as a free group (Remeslennikov, Sibirsk. Mat. Zh., 30 (1989), 193–197). The subclass, elementarily free groups, are those that have the same elementary theory as a free group.
The von Neumann–Tits line runs between the two coasts: all the groups below it contain subgroups, all above do not.
But there remain expanses of wilderness where, thanks to Higman’s Embedding Theorem and the like, one might say “Here there be dragons!” And, as will become apparent in this blog, these dragons do not keep themselves to the distant wildernesses. They venture into the pastures of non-positive curvature, showing themselves in subgroup structure.