Dictionary metric on group

Vocabulary metric is a way to specify distances on a finitely generated group .

Content

Design

If the final system of generators is selected and fixed $F$ ${\ displaystyle F}$ in a finitely born group $\Gamma$ ${\ displaystyle \ Gamma}$ , then the distance between the elements $g$ ${\ displaystyle g}$ and $h$ ${\ displaystyle h}$ Is the smallest number of generators and inverses to them, into the product of which the quotient is decomposed $g^{-1}h$ ${\ displaystyle g ^ {- 1} h}$ .

Properties

The dictionary metric is left-invariant; that is, left multiplication is preserved by a fixed group element.
- For non-Abelian groups, generally speaking, it is not right-invariant.
The dictionary metric coincides with the distance in the Cayley graph for the same system of generators.
The vocabulary metric is not preserved when replacing the system of generators, but it changes quasi-isometrically (in this case, it is the same as in a bilipschitz manner). That is, for some constants $C_{1},C_{2}$ ${\ displaystyle C_ {1}, C_ {2}}$ takes place:
$\forall g,h\in G\quad {\frac {1}{C_{1}}}d_{2}(g,h)\leq d_{1}(g,h)\leq C_{2}d_{2}(g,h).$ ${\ displaystyle \ forall g, h \ in G \ quad {\ frac {1} {C_ {1}}} d_ {2} (g, h) \ leq d_ {1} (g, h) \ leq C_ { 2} d_ {2} (g, h).}$ .

In particular, this makes it possible to apply geometric concepts to the group using the dictionary metric, which are preserved under quasi-isometry. For example, to talk about the degree of growth of the group (polynomial, exponential, intermediate) and its hyperbolicity .

Variations and generalizations

In a similar way, the dictionary metric can be built on an arbitrary group (not necessarily finitely generated), while it becomes necessary to take an infinite system of generators and many of the described properties cease to be fulfilled.

Links

JW Cannon, Geometric group theory, in Handbook of geometric topology pages 261--305, North-Holland, Amsterdam, 2002, ISBN 0-444-82432-4