metric property in a group

Question

Can we define a metric function on a group $G$? Please give examples other than $\mathbb{R}$. Actually most groups have elements in discrete manner. It sounds vague but I can't be more precise.

Answer 1

As I mentioned in my comment, discrete spaces can have metrics, so the integers, with the metric induced from $\mathbb{R}$, are an example.

However, I don't think that will satisfy you. Here is a better one (this is actually the simplest example of a compact Lie group). Consider the group $U(1)$, which you can think of as complex numbers of unit modulus, with multiplication as the group operation: $$ e^{i\theta}\cdot e^{i\theta'} = e^{i(\theta + \theta')} ~. $$ Obviously the group elements correspond to points on the unit circle in the complex plane. Therefore this group has an obvious metric: the one induced from the standard metric on $\mathbb{C} \cong \mathbb{R}^2$.

Answer 2

Word Metric: There is something called the word metric for groups, which is a key tool in geometric group theory.

Consider a group given by some generators, so $G=\langle X\rangle$. Then you can draw the Cayley graph of $G$ with respect to these generators. Take two elements $g, h\in G$, then define $d(g, h)$ to be the minimum distance in the Cayley graph from the vertex labelled $gh^{-1}$ to the vertex labelled with the identity element. For more details, see Wikipedia.

For an example of why the word metric can be very powerful, look up hyperbolic groups (also known as word-Hyperoblic or Gromov-Hyperbolic groups). These are groups which are be defined by saying that they have a word metric which possesses certain "hyperbolic-like" properties (there are other, equivalent definitions though). Hyperbolic groups have soluble word problem and conjugacy problem, and recently it has been shown that they have soluble isomorphism problem (all three problems are insoluble in general). Indeed, Hyperbolic groups have the "fastest possible" word problem, which kinda follows from the "hyperbolic" bit, because area in a hyperbolic space is $\pi r$ rather than $\pi r^2$, and the concept of area in the word metric is very much related to the word problem ("area" relates to a closed cycle in the graph, and a word defines the trivial element if it defines a closed cycle through the origin).

Another interesting class of groups defined using the word metric are automatic groups.

Generalising: The key point with the word metric is that the group $G$ acts (in a nice way) on the Cayley graph. If a group acts (in a nice way) on a metric space then you can define a metric on the group using the metric on the space. Again, this is very fruitful. An excellent text for this is the book of Bridson-Haefliger, Metric Spaces of Non-Positive Curvature, with Chapter I.8 being where the stuff relevant to here really starts.

Answer 3

Let $G=(\Bbb Z/2\Bbb Z)^\omega$, the direct product of countably infinitely many copies of $\Bbb Z/2\Bbb Z$, the cyclic group of order $2$. Elements of $G$ can be viewed as infinite sequences of zeroes and ones. Give $\Bbb Z/2\Bbb Z$ the discrete topology and $G$ the usual product topology; then $G$ is a Cantor set, so it’s a compact space with no isolated points. For $x=\langle x_n:n\in\omega\rangle,y=\langle y_n:n\in\omega\rangle\in G$ with $x\ne y$ define $$\delta(x,y)=\min\{n\in\omega:x_n\ne y_n\}\;,$$ and define

$$d:G\times G\to\Bbb R:\langle x,y\rangle\mapsto\begin{cases} 0,&\text{if }x=y\\ 2^{-\delta(x,y)},&\text{if }x\ne y\;. \end{cases}$$

Then $d$ is a metric on $G$ compatible with the group operation. (It is in fact an ultrametric or non-Archimedean metric: it satisfies the strong triangle inequality $$d(x,y)\le\max\{d(x,z),d(z,y)\}$$ for all $x,y,z\in G$.)

$G$ is essentially the additive group of $2$-adic integers equipped with its usual metric. For any prime $p$ the $p$-adic integers furnish a good example; these lecture notes discuss the ring (hence group) $\Bbb Z_p$ of $p$-adic integers, defining an ultrametric $d_p$ on it; the relevant discussion concludes at the top of page $5$. Each of the metric spaces $\langle\Bbb Z_p,d_p\rangle$ is homeomorphic to the Cantor set. $\Bbb Z_p$ can be extended to the field of $p$-adic numbers, which can be naturally equipped with a metric extending $d_p$; the resulting space is homeomorphic to the Cantor set minus one point or, equivalently, to the discrete union of countably infinitely many copies of the Cantor set.

Of course one can also start with any metric group $\langle G,d\rangle$ and construct the metric group $\langle G^\omega,\bar d\rangle$ by defining

$$\bar d(x,y)=\sum_{n\ge 0}\frac{d(x_n,y_n)}{2^n\big(1+d(x_n,y_n)\big)}$$

for each $x=\langle x_n:n\in\omega\rangle,y=\langle y_n:n\in\omega\rangle\in G^\omega$.