Random subgroups of a free group

Rndom sugroups of free group Frédérique Bssino LIPN - Lortoire d Informtique de Pris Nord, Université Pris 13 - CNRS Joint work with Armndo Mrtino, Cyril Nicud, Enric Ventur et Pscl Weil LIX My, 2015

Introduction Any group is isomorphic to quotient group of some free group. Study of lgeric properties of free groups y comintoril methods Grphicl representtion of sugroups : Stllings grphs Comintoril interprettion of prmeters or properties like the rnk, mlnormlity, Whitehed minimlity,... Quntittive study of finitely generted sugroups of free group nd nlysis of relted lgorithms

I. Free Group

Free group : definition A group F is free if there is suset A of F such tht ny element of F cn e uniquely written s finite product of elements of A nd their inverses. The crdinlity of A is the rnk of the free group. Aprt from the existence of inverses no other reltion exists etween the genertors of free group. Bsic properties The sugroups of free group re free (Nielsen-Schreier Theorem). A free group with finite rnk contins sugroups with ny countle rnk.

Free groups nd reduced words Let A e finite lphet nd F = F(A) e the free group over A. The elements of F(A) re uniquely represented y the reduced words over A A 1 where A 1 = { 1 A}, A word is reduced if it does not contin fctors of the form 1 Exmples : 1 1 1 is reduced, 1 1 cc 1 is not reduced Reduction of word : replce in ny order ll occurrences of 1 y the empty wordǫ. Exmple : 1 1 cc 1 = 1 cc 1 = cc 1

Finitely generted sugroups We re interested in finitely generted free sugroups, i.e., otined from finite set of genertors. Finitely generted free sugroups cn e represented in unique wy y finite grph clled its Stllings grph (Stllings 1983). This description is very useful, some properties of the sugroup cn e directly otined from its grph representtion. A 1st gol To study lgeric properties of finitely generted sugroups of free group with comintoril methods.

Stllings foldings Let Y = { 1 1, 2 1, 3 1 1 }. Gol Build directed grph representing the free sugroup generted y Y First step Build directed cycle leled with 1 1 the first element of Y 1 1 i

Stllings foldings Second step Build from the sme vertex i directed cycle leled with 2 1 the second element of Y. 1 1 1 i

Stllings foldings Third step Build from the sme vertex i directed cycle leled with 3 1 1 the third nd lst element of Y. 1 1 1 i 1 1

Stllings foldings Forml inverses Reverse ll edges leled y 1 re nd replce their lel y. i

Stllings foldings Foldings to otin determinism nd codeterminism Apply s mny times s possile the following rules of merging (or folding) : The result does not depend on the order in which the trnsformtions re performed.

Stllings foldings - 1st folding i i

Stllings foldings - 2nd folding i i

Stllings foldings - 3rd folding A i B B A i

Stllings foldings - 4th folding i i

Stllings foldings - Lst folding nd Stllings grph i The Stllings grph representing the free sugroup generted y Y = { 1 1, 2 1, 3 1 1 }. i

Stllings grphs : definition The grph (with distinguished vertex i) otined is Stllings grph. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. it is connected ll ut the distinguished stte i hve degree t lest two Unicity of the representtion A Stllings grph represents in unique wy finitely generted sugroup of the free group generted y the lphet of the lels.

Stllings grphs exmples of use One cn check whether (reduced) word elongs the sugroup or not. Check if there exists cycle leled y the word eginning in i One cn compute sis nd the rnk of the sugroup rnk = E ( V 1) To otin sis, choose spnning tree of the Stllings grph. Ech edge e tht is not in the tree corresponds to genertor of the se : the lel of cycle eginning in i using e nd edges in the spnning tree. One cn check whether the sugroup hs finite index or not. All letters ct like permuttions on the set of vertices

Exmple for the rnk The Stllings grph of the sugroup genrted y Y = { 1 1, 2 1, 3 1 1 } : Therefore{ 2 1, 1 1 } is sis of the sugroup nd the rnk is 2. i

Stllings grphs lgorithmic point of view Stlling foldings cn e computed in O(n log n) where n is the totl length of the genertors. The lgorithm due Touikn (2006) mkes use of Union nd Find. The intersection (resp. union) of two sugroups cn e computed in time nd spce O(n 1 n 2 ) where n 1 (resp. n 2 ) is the size (here the numer of vertices) of the first (resp. second) Stllings grph.

II. Distriutions on Sugroups

A grph-sed distriution on sugroups A rndom sugroup is given y choosing uniformly t rndom Stllings grph of size n Studied y Bssino, Nicud, Weil (2008, 2013, 2015) Wht does the Stllings grph of such rndom sugroup look like? FIGURE: A rndom sugroup with 200 vertices for the grph-sed distriution (The lphet is of size 2).

The clssicl word-sed distriution on sugroups A rndom sugroup is given y choosing rndomly nd uniformly k genertors of length t most n, where k is fixed Studied y Gromov (1987), Arzhntsev nd Ol shnskiǐ (1996), Jitsukw (2002),... Wht does the Stllings grph of such rndom sugroup look like? FIGURE: A rndom sugroup for the word-sed distriution with 5 words of lengths t most 40 (The lphet is of size 2.)

A word-sed distriution (few genertors) Fix the numer k of genertors nd the mximl length n of ech genertor. Consider the uniform distriution over the k-tuples of reduced words of length t most n. Let R n the numer of reduced words of length n, R n = 2r(2r 1) n 1 The length of word in rndom k-tuple is ner to n.

A word-sed distriution (few genertors) Length, prefixes nd suffixes Let 0 < α < 1. A reduced word in R n hs length greter thnαn with proility tht tends towrd 1 when n tends towrd+. Let 0 < β < α/2. A k-uple of reduced words of R n is such tht the prefixes of lengthβn of ll words nd their inverses re pirwise distinct with proility tht tends towrd 1 when n tends towrd+. Consequence Ech of the k reduced words hs n outer loop of length t lest n(α 2β) with proility tht tends to 1 when n tends to+.

A grph-sed distriution : Proilistic results Theorem (Bssino, Nicud, Weil 2008) The proility for rndom r-tuple of prtil injections of size n to form Stllings grph tends towrd 1 when n tends towrd +. Stllings grph It is deterministic nd co-deterministic : ech letter cts like prtil injection on the set of sttes. The proof it is connected ll ut the distinguished stte i hve degree t lest two is study of prtil injections siclly uses the sddle-point method

A grph-sed distriution : Prtil injections A prtil injection cn e seen s set of cycles nd of non-empty sequences. Set(Cycle or non-empty Sequences) With the symolic method : I(z) = n 0 I n n! zn = exp ( log With the sddle point method : I n n! e 2 π e2 1 1 z + z ) = 1 1 z 1 z ez/(1 z) 1 2 n n 1 4

Connectedness Theorem The proility for r prtil injections of size n to form connected grph is ( ) p n = 1 2r 1 n r 1 + o n r 1 Proof Let J(z) = n>0 j nz n = n>0 Ir nz n /n!. Then 1+J(z) = exp(c(z)) nd C(z) = log(1+j(z)). From Bender theorem (1974) it is enough to check tht j n = o(j n 1 ) nd tht for some s 1, n s k=s j kj n k = O(j n s ), to otin tht c n = j n ( 1 2r n r 1 + o ( 1 n r 1 ))

Vertices with zero or one outgoing or ingoing edge If x is vertex with 0 or 1 edge, then x must e isolted for r 1 injections nd n endpoint for the remining injection. The proility it is isolted for n injection is I n 1 I n, which is smller thn 1 n. Let I n,k e the numer of size-n injections hving k sequences, nd let I(z, u) e the ivrite generting function defined y : ( ( )) zu 1 I(z, u) = exp 1 z + log = 1 ( ) zu 1 z 1 z exp 1 z Using the sddle point theorem we otin tht the expected numer of sequences is 1 n nd tht the proility tht given vertex is n endpoint is in O( 1 n ).

Trimness Therefore A given vertex hs degree 0 or 1 with proility O(n r+1/2 ), there is such vertex with proility O(n r+3/2 ) with proility t lesto(n 1/2 ) the grph hs no such vertex.

IV. How to compre the two distriutions

Méthod A property P is generic for(x n ) when the proility for n element of X n to stisfy P tends towrd 1 when n tends towrd. A property P is negligile for(x n ) when the proility for n element of X n to stisfy P tends towrd O when n tends towrd. In the following, we present generic or negligile lgeric properties for ech distriution.

Experimentl results FIGURE: On the left, rndom sugroup for the word-sed distriution with 5 words of lengths t most 40. On the right, rndom sugroup with 200 vertices for the grph-sed distriution (The lphet is of size 2).

Rnk One cn compute the rnk of finitely generted sugroup from its Stllings grph rnk = E ( V 1) In the word sed distriution (k words of mximl length n), the verge rnk is k In the grph sed distriution the verge rnk is ( A 1)n A n+1.

Mlnormlity A sugroup H of G is norml when for ny g G, g 1 Hg = H. A sugroup is mlnorml when for ny g / H, g 1 Hg H = 1. Theorem (comintoril chrcteriztion) A sugroup of free group is non-mlnorml if nd only, in its Stllings grph, if there exists two vertices x y nd non-empty reduced word u, such tht u is the lel of loop on x nd of loop on y.

Mlnormlity Theorem For the word-sed distriution, mlnormlity is generic, ut it is negligile for the grph-sed. Proof The proof in the word-sed distriution is due to Jitsukw (2002). Bsiclly loops re long enough to e distinct with high proility. The proility tht prtil injection contins t most one cycle nd tht the length of this cycle is 1 is e n.

Finite presenttion The ide is to quotient the free group y norml finitely generted sugroup. Let E e n ritrry suset, nd N(E) e its norml closure, tht is the smllest norml sugroup contining E. Equivlently ech word x of E ecomes reltor x = 1. In the word-sed distriution genericlly the quotient sugroup is infinite (Jitsukw, 2002). But in the grph-sed distriution, the quotient group is genericlly trivil.

Finite presenttion Theorem Genericlly the gcd of the lengths of the cycles of prtil injection of size n is 1. Theorem Genericlly the gcd of the lengths of the cycles of permuttion of size n is 1. Permuttion prt of n injection Genericlly the permuttion prt of size n injection is greter thn n 1/3 nd the gcd of the length of the cycles is 1.

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