Length function: Difference between revisions

Revision as of 19:57, 20 April 2024

Revision as of 05:43, 14 July 2024

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{{unreferenced|date=November 2008}}

In the [[mathematics|mathematical]] field of [[geometric group theory]], a '''length function''' is a [[function (mathematics)|function]] that assigns a number to each element of a [[group (mathematics)|group]].

==Definition==

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,\\