THE GEOMETRY MONOID OF AN IDENTITY

Transcription

1 THE GEOMETRY MONOID OF AN IDENTITY Parick DEHORNOY Universié decaen Main idea: For each algebraic ideniy I, (more generally, for each family of algebraic ideniy, acually for each equaional variey), here exiss a specific monoid M I ha describes he geomery of I. Sudying M I wih convenien algebraic ools leads in good cases o resuls abou I and I-sysems (= hose algebraic sysems ha saisfy I, ypically: - solving he word problem, - consrucing free I-sysems. Applies a leas o - x(yz) =(zy)z (Thompson, MacLane, Sasheff); - x(yz) =(xy)(xz) ( braid applicaions); - x(yz) =(xy)(yz) (new)... 1

2 Free I-sysems: Suppose I is an algebraic ideniy involving one binary operaion, for insance x (y z) =(x y) (y z). (I) Fix a se of variables X; Le T X be he se of all erms consruced from X and a binary operaor; Le I be he congruence on (he absoluely free algebra) T X generaed by he insances of I, i.e., he pairs ( 1 ( 2 3 ), ( 1 2 ) ( 2 3 )). Fac: T X / I is a free I-sysem based on X. Wha does applying I oaerm mean? Ieraively replacing some suberm of which has he form 1 ( 2 3 ) wih he corresponding erm ( 1 2 ) ( 2 3 ), or conversely: depends on orienaion and on posiion. 2

3 The operaors I + a : Fix an address sysem in erms : view hem as binary rees and specify a suberm by describing he pah from he roo: he -h suberm of Definiion: I a + is he (parial) operaor on T X ha maps o iff he -h suberm of canbeexpressed as 1 ( 2 3 ) and is obained from by replacing his suberm wih he corresponding ( 1 2 ) ( 2 3 ) (= applying I o a ); wrie I for he inverse of I a +. 3

4 The geomery monoid of I: Definiion: The geomery monoid M I of I is he monoid generaed by all operaors I + a and I. Fac: Two erms, are I -equivalen iff some elemen of M I maps o : =()w, where w is a finie sequence of signed addresses (describing how o ransform o using I). Quesion: HowouseM I? In paricular: Can he sudy of M I solve he word problem of I? difficul, because (i) M I is no a group, (ii) here is no uniform connecion beween M I, which acs on erms, and erms hemselves; soluion when (i) M I can be replaced wih a group, (ii) M I conains copies of he erms (in some sense...) 4

5 The group G I : Principle: Guess a presenaion of M I, hen inroduce he group G I defined by his presenaion: hopefully: G I ressembles M I enough. Geomery relaions in M I : Example: I 10β + I+ = I + I 01β + I+ 10β, or simply 10β 01β 10β. I + I + 10β I + 01β I+ 10β I + 5

6 Definiion: The group G I is he group {0, 1} ; R I, wih R I he lis of all relaions: 0β 1γ = 1γ 0β, 0β = 00β, 10β = 01β 10β, 11β = 11β, 1 0 = 1. How o sudy such a group? G I is he group of fracions of a monoid which admis (righ) leas common muliples (proving his requires specific algebraic ools, mainly word reversing, reminiscen of Garside s analysis of he braid groups. 6

7 The blueprin of a erm: How o connec he monoid M I and he group G I? HowouseG I for sudying I? Fac: For each in T {x},wehave p+1 I x p for p large enough. Proof: For = x, equaliy. For = 1 2 : x p+1 I 1 x p for p large enough. I 1 ( 2 x p 1 ) I ( 1 2 ) ( 2 x p 1 ) I ( 1 2 ) x p = x p Some elemen of M I, depending on, muswiness for his erm equivalence: use his elemen, or, raher, is copy in G I,asheblueprin of. Definiion: For in T {x},heblueprin of is he elemen χ of G I inducively defined by χ x = 1, and χ = χ 1 sh 1 (χ 2 ) sh 1 (χ 1 1 ) for = 1 2, where sh 1 : 1 for each address. 7

8 By consrucion, we have (x p+1 )χ = x p for p large enough: hus χ, which lives in G I, describes how o consruc from scrach using I. ( no every ideniy is eligible). Now, use χ as a synacic counerpar o : Assume I, hence =()w for some w. Then x p+1 x p+1 χ x p χ x p sh 0 (w) x p where sh 0 : 0: if we guessed he relaions correcly, we should have χ χ sh 0 (w); If his is rue, his mus be checkable by a direc compuaion. This is rue. 8

9 Proposiion: (soluion o he word problem) For, in T {x}, he following are equivalen: (i) We have I ; (ii) In he group G I, he elemen χ 1 χ belongs o he subgroup generaed by he elemens 0. For which ideniies does his approach work? associaiviy Thompson s group F ; self-disribuiviy an exension of Arin s braid group B ; he curren ideniy x(yz) =(xy)(xz)... In each case: specific algebraic sudy (he groups are very differen). References P. Dehornoy; Braids and Self-Disribuiviy; Progress in Mah. vol. 192, Birkhäuser (2000). Preprins: hp:// dehornoy/ 9