Why a group should have a shape
By now you have learned to do geometry with nothing but a distance: a length space measures curves, a geodesic space realizes shortest paths, and Gromov-Hausdorff distance compares whole spaces at once. The radical idea of geometric group theory is to point this machinery at an object that seems purely algebraic — a finitely generated group G — and insist that it, too, is a geometric space. A group is usually handed to you as a presentation: generators and relations, multiplication, inverses. Nothing in that data looks like a metric. The bridge is to let the group act on itself and read distances off the action.
Recall from Vol I how the fundamental group of a space already carried hidden geometry: loops up to homotopy, with concatenation as multiplication. Here we do something complementary. Instead of extracting a group from a space, we manufacture a space from a group, then ask which of its geometric features are real and which are artifacts of our choices. The honest punchline, which the rest of this guide earns, is that almost every fine detail is an artifact — but the coarse large-scale shape is canonical, and that coarse shape remembers a startling amount about the algebra.
The Cayley graph and the word metric
Fix a finite generating set S for G, and assume S is symmetric (if s is in S then so is s^{-1}). The Cayley graph Cay(G, S) is a graph whose vertices ARE the elements of G: every group element is a dot. Draw an edge from g to gs for each generator s in S. So you stand at g and the generators are your menu of single steps; taking step s lands you at gs. The result is a connected graph — connected precisely because S generates G, so any element is some product of generators, i.e. some walk from the identity.
Now declare every edge to have length 1 and you get a word metric: the distance d_S(g, h) is the fewest edges in any path from g to h, equivalently the length of the shortest word in the generators spelling g^{-1} h. The distance from the identity e to g, written |g|_S, is the word length of g — the minimum number of generator-steps to reach it. This is a genuine, integer-valued metric, and the graph with this metric is a geodesic space: a shortest edge-path between two vertices literally realizes the distance.
Spelled out: the vertices of Cay(G, S) are the elements g of G, and for each generator s in S there is a unit-length edge joining g to gs. The word length is then |g|_S = min { n : g = s_1 s_2 ... s_n with each s_i in S }, and the word metric is d_S(g, h) = |g^{-1} h|_S. Two small pictures fix the idea. For G = Z with S = {+1, -1}, the Cayley graph is just the integer number line ...-2-1-0-1-2..., and the word metric is ordinary |m - n|: the simplest geometry imaginable.
Going further: for the free group F_2 on two generators a, b, the Cayley graph is the infinite 4-valent tree — at every vertex four edges (a, a^{-1}, b, b^{-1}) sprout, and there are no loops because there are no relations to close them up. That tree is already, secretly, the prototype of a Gromov-hyperbolic space: thin triangles, exponential branching, a boundary at infinity.
The catch: the metric depends on your generators
Here is the embarrassment that nearly sinks the whole program. The word metric is not an invariant of G alone — it depends on the generating set S you chose. Take G = Z again but with S = {+2, -2, +3, -3}. Now you can reach 5 in two steps (3 then 2), so |5| drops to 2 instead of 5; the entire metric is rescaled and reshaped. Worse, the Cayley graph itself changes: different S give graphs that are not even isomorphic. If geometry is supposed to capture the group, why does it wobble the moment we change a cosmetic choice?
The rescue is to stop demanding that distances match exactly and accept distances that match up to bounded multiplicative and additive error. Compare the two metrics on Z. Any single step of size 2 or 3 costs at most 3 units in the standard {+1, -1} metric, and any standard unit step costs at most 1 in the coarse metric. So the two metrics never disagree by more than a fixed factor and a fixed constant. They look identical from far away, like two road maps drawn at slightly different scales. That bounded discrepancy is exactly what we will promote to the central equivalence.
Quasi-isometry: equality up to bounded error
A map f: X -> Y between metric spaces is a quasi-isometric embedding if there are constants L >= 1 and C >= 0 so that for all points, (1/L) d_X(x, x') - C <= d_Y(f(x), f(x')) <= L d_X(x, x') + C. The map need not be continuous, injective, or surjective; it only has to preserve distances up to stretching by L and slack C. It is a full quasi-isometry if, in addition, every point of Y lies within distance C of the image f(X) — the image is C-dense, so f does not miss whole regions. Two spaces are quasi-isometric when such an f exists, and this is an equivalence relation on metric spaces.
f: X -> Y is a (L, C)-quasi-isometric embedding ( L >= 1, C >= 0 ) iff
(1/L) d_X(x, x') - C <= d_Y( f(x), f(x') ) <= L d_X(x, x') + C
... and a quasi-isometry iff also every y in Y has d_Y( y, f(X) ) <= C.
Non-example you must keep straight:
Z is NOT quasi-isometric to Z^2 ( their growth differs )
Z IS quasi-isometric to R ( the inclusion has L=1, C=1 )Quasi-isometry is deliberately coarse. The additive C blinds it to any bounded local detail: a space and the same space with a single point smeared into a small blob of diameter 5 are quasi-isometric, because the difference is swallowed by C. The multiplicative L blinds it to the scale. This is why Z with generators {+1, -1} and Z with {+2, -2, +3, -3} are quasi-isometric: the identity map on Z is a quasi-isometry between the two word metrics. The fine wobble we worried about is invisible at this resolution.
The payoff: a group's geometry is well defined
Now collect the reward. For a finitely generated group G, any two finite generating sets S and S' give word metrics that are quasi-isometric via the identity map (the argument is exactly the Z example: each S-generator is a bounded S'-word and vice versa, bounding L; you may take C = 0). Therefore the quasi-isometry class of Cay(G, S) does NOT depend on S. We may finally speak of the large-scale geometry of G, an honest invariant of the group. Anything you prove that is invariant under quasi-isometry is a genuine theorem about G, free of the arbitrary generating set.
What survives this coarsening is exactly the deep stuff. The growth type of a group — whether the number of elements of word length at most n grows polynomially, exponentially, or in between — is a quasi-isometry invariant, and Gromov's celebrated theorem says polynomial growth happens exactly for the virtually nilpotent groups. Being a hyperbolic group (the Cayley graph is Gromov-hyperbolic, with uniformly thin triangles) is a quasi-isometry invariant too, as is the number of ends and many other features. These are theorems where geometry feeds genuinely new facts back into algebra.
One honest caveat before the next guide. Quasi-isometry is coarser than algebra: quasi-isometric groups need not be isomorphic, nor even commensurable, and deciding whether two given groups are quasi-isometric is hard and often open. Conversely, the link from a group's geometry to a space it acts on is supplied by the Švarc-Milnor lemma — sometimes called the fundamental observation of geometric group theory — which the next guide develops: when G acts nicely (properly and cocompactly by isometries) on a geodesic space X, then G with its word metric is quasi-isometric to X. That is how an abstract group inherits the geometry of the space it lives on, and how growth re-enters as the volume growth of that space.