Length function: Difference between revisions

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LetIn <math>G</math>the be[[mathematics|mathematical]] afield of [[inverse|geometric group theory]]., Aa '''length function''' on <math>G</math> is a [[codomainfunction (mathematics)|function]] that <math>L\colonassigns Ga number \to \mathbb{R}^+</math>each satisfying:element of a [[group (mathematics)|group]].

==Definition==

<table cellpadding="0" align="center" width="100%"> <tr valign="middle"> <td nowrap="nowrap" align="right">

A '''length function''' ''L''&nbsp;:&nbsp;''G''&nbsp;&rarr;&nbsp;'''R'''⁺ on a [[group (mathematics)|group]] ''G'' is a function satisfying:<ref>{{citation

| last = Lyndon | first = Roger C.

| doi = 10.7146/math.scand.a-10684

| journal = Mathematica Scandinavica

| jstor = 24489388

| mr = 163947

| pages = 209–234

| title = Length functions in groups

| volume = 12

| year = 1963}}</ref><ref>{{citation

| last = Harrison | first = Nancy

| doi = 10.2307/1996098

| journal = Transactions of the American Mathematical Society

| mr = 308283

| pages = 77–106

| title = Real length functions in groups

| volume = 174

| year = 1972}}</ref><ref>{{citation

| last = Chiswell | first = I. M.

| doi = 10.1017/S0305004100053093

| issue = 3

| journal = Mathematical Proceedings of the Cambridge Philosophical Society

| mr = 427480

| pages = 451–463

| title = Abstract length functions in groups

| volume = 80

| year = 1976}}</ref>

:<math>\begin{align}L(e) &= 0,\\

:<math> L(e)</math></td> <td width="10" align="center" nowrap="nowrap">

L(g^{-1}) &= L(g)\\

L(g_1 g_2) &\leq L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G.

\end{align}</math>

Compare with the [[axiom]]s for a [[metric (mathematics)|metric]] and a [[filtered algebra]].

:<math> =</math></td> <td align="left" nowrap="nowrap">

==Word metric==

:<math> 0,</math></td> <td class="eqno" width="10" align="right"> </td> </tr> <tr valign="middle"> <td nowrap="nowrap" align="right">

{{main|Word metric}}

An important example of a length is the [[word metric]]: given a [[presentation of a group]] by generators and relations, the length of an element is the length of the shortest word expressing it.

[[Coxeter group]]s (including the [[symmetric group]]) have combinatorialcombinatorially important length functions, forusing whichthe simple reflections as generators (thus each simple reflection has length &nbsp;1). See also: [[length of a Weyl group element]].

:<math> L(g)</math></td> <td width="10" align="center" nowrap="nowrap">

A [[longest element of a Coxeter group]] is both important and unique up to conjugation (up to different choice of simple reflections).

:<math> =</math></td> <td align="left" nowrap="nowrap">

==Properties==

:<math> L(g^{-1}), \quad\forall g \in G,</math></td> <td class="eqno" width="10" align="right"> </td> </tr> <tr valign="middle"> <td nowrap="nowrap" align="right">

A group with a length function does ''not'' form a [[filtered group]], meaning that the [[sublevel set]]s <math>S_i := \{g \mid L(g) \leq i\}</math> do not form [[subgroup]]s in general.

However, the [[group ring|group algebra]] of a group with a length functions forms a [[filtered algebra]]: the axiom <math>L(gh) \leq L(g)+L(h)</math> corresponds to the filtration axiom.

:<math> L(g_1 g_2)</math></td> <td width="10" align="center" nowrap="nowrap">

==References==

:<math> \leq</math></td> <td align="left" nowrap="nowrap">

{{reflist}}

{{planetmathPlanetMath attribution|id=4365|title=Length function}}

:<math> L(g_1) + L(g_2), \quad\forall g_1, g_2 \in G.</math></td> <td class="eqno" width="10" align="right"> </td> </tr> </table>

[[Coxeter group]]s (including the [[symmetric group]]) have combinatorial important length functions, for which each simple reflection has length 1.

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{{planetmath|id=4365|title=Length function}}

[[Category:Group theory]]

[[Category:Geometric group theory]]