Ideal Whitehead Graphs in Out(F r ) II: The Complete Graph in Each Rank

Transcription

1 Idel Whitehed Grphs in Out(F r ) II: The Complete Grph in Eh Rnk Ctherine Pfff Astrt We show how to onstrut, for eh r, n geometri, fully irreduile φ Out(F r ) whose idel Whitehed grph is the omplete grph on r 1 verties. This pper is the seond in series of three where we show tht preisely eighteen of the twenty-one onneted, simpliil, five-vertex grphs re idel Whitehed grphs of fully irreduile φ Out(F ). The result is first step to n Out(F r ) version of the Msur-Smillie theorem proving preisely whih index lists rise from singulr mesured folitions for pseudo-anosov mpping lsses. In this pper we dditionlly give method for finding periodi Nielsen pths nd prove riterion for identifying representtives of geometri, fully irreduile φ Out(F r ) 1 Introdution In [MS9] Msur nd Smillie list preisely whih singulrity index lists rise from the pir of invrint folitions for pseudo-anosov mpping lss. The index lists were signifint in their strtifition of the spe of qudrti differentils into strt invrint under the Teihmuller flow. Severl ppers studying the strtifition inlude [KZ0], [Ln04], [Ln05], nd [Zor10]. While Out(F r ) index theory hs een developed in ppers suh s [GJLL98], [GL95], [CH], nd [CH1], this is the first on index reliztion. Our serh for idel Whitehed grphs rising from fully irreduile free group outer utomorphisms is motivted y our gol of determining the Out(F r )-version of the Msur-Smillie theorem. An idel Whitehed grph (see Setion ) is finer invrint thn singulrity index list nd enodes informtion out the ttrting lmintion for fully irreduile outer utomorphism. We onstrut, for eh r, n geometri, fully irreduile φ Out(F r ) whose idel Whitehed grph is the omplete (r 1)-vertex grph. We onsequently prove: Theorem. Let C r denote the omplete (r 1)-vertex grph. For eh r, there exists n geometri, fully irreduile φ Out(F r ) suh tht C r is the idel Whitehed grph IW(φ) for φ. Tht the (r 1)-vertex omplete grph ours s n idel Whitehed grph in eh rnk is oth nonovious nd signifint. First, the omplete grph hs no ut verties. This phenomen holds signifine for other versions of Whitehed grphs (see, for exmple, [Cs10], [MM10], [St99]). Additionlly, y [HM11], ut verties in n idel Whitehed grph hve implitions out periodi Nielsen pths. Seond, the (r 1)-vertex omplete grph is onneted grph yielding the index sum r. This sum is s lose s possile to tht of 1 r, hieved y geometris (fully irreduiles indued y pseudo Anosov surfe homeomorphisms), without eing hieved y geometri outer utomorphism. As in [Pf1], we denote the set of onneted (r 1)-vertex simpliil grphs y PI (r;( r)). Third, the existene of suh omplited idel Whitehed grphs highlights signifint depth within Out(F r ) theory, 1

2 eyond tht of mpping lss groups, s the idel Whitehed grph of ny pseudo-anosov is simply disjoint union of irles. To show tht our exmples indeed represent fully irreduiles, we prove folk lemm, Proposition 4.1, the Full Irreduiility Criterion (FIC). [Kp1] gives nother riterion inspired y our FIC. Proposition. 4.1 (The Full Irreduiility Criterion) Let g e trin trk representing n outer utomorphism φ Out(F r ) suh tht (I) g hs no periodi Nielsen pths, (II) the trnsition mtrix for g is Perron-Froenius, nd (III) ll lol Whitehed grphs LW(x; g) for g re onneted. Then φ is fully irreduile. We ddress three issues when pplying the riterion to trin trk representtive g : Γ Γ. First, we must verify tht g hs no periodi Nielsen pths. Rell [BH9] tht, for trin trk mp g : Γ Γ, nontrivil pth ρ in Γ is lled periodi Nielsen Pth (pnp) if, for some k, g k (ρ) ρ rel endpoints. We ensure our prtiulr representtives re pnp-free using Nielsen pth prevention sequene, see Lemm 5.5. Proposition 5. provides method for identifying pnp s of trin trk mps idelly deomposed in the sense of [Pf1] ( method of different nture n e found in [Tur94]). This proedure n lso e used to prove tht n idelly deomposed representtive hs no pnp s. Seond, we need tht the lol Whitehed grph LW(x; g) t eh vertex x of Γ is onneted, ondition stisfied in our se y the idel Whitehed grph eing onneted. Third, the FIC inludes ondition on the trnsition mtrix (s defined in [BH9]) for g, stisfied when some g n mps eh edge of Γ over eh other edge of Γ. We use swith sequene to ensure this property is stisfied (Lemms.11 nd.1). Finlly, to ensure our representtives tully hve the orret idel Whitehed grphs, the representtives re onstruted using pths in the lmintion trin trk strutures of [Pf1]. The pths orrespond to Dehn twist utomosphisms x xw nd onstrut the lmintion, s do the Dehn twists in [Pen88]. In [CP10] Cly nd Pettet lso use Dehn twists to onstrut fully irreduiles, ut fous on sugroups generted y powers of two Dehn twists for two filling yli splittings. Further onstrution methods for fully irreduile outer utomorphisms n e found in [KL10] nd [Hm09]. In [Pf1] we gve, for eh r, exmples of onneted, simpliil (r 1)-vertex grphs tht re not the idel Whitehed grph of ny fully irreduile outer utomorphism φ Out(F r ). In [Pf1] we will finish our proof tht preisely eighteen of the twenty-one onneted, simpliil, five-vertex grphs re idel Whitehed grphs of fully irreduile φ Out(F ). The results of this pper re used for proving the theorem in [Pf1], ut lso mke progress in seond diretion, s we prove existene of the omplete grph in eh rnk insted of fousing exlusively on rnk-three, s we will in [Pf1]. Aknowledgements The uthor would like to thnk Lee Mosher, s lwys, for truly invlule onverstions, Arnud Hilion for dvie, nd Mrtin Lustig for ontinued interest in her work. She would lso like to thnk Mihel Hndel for his reommendtion on how to omplete the Full Irreduiility Criterion proof. Finlly, she would like to thnk the referee for very thoughtful nd helpful referee report. She extends her grtitude to Brd College t Simon s Rok nd the CRM for their hospitlity. Preliminry definitions nd nottion We ontinue with the introdution s nottion. Additionlly, unless otherwise stted, we ssume throughout this doument tht outer utomorphism representtives re trin trk

3 (tt) representtives in the sense of [BH9]. For rnk r free group F r, FI r will denote the set of the fully irreduile elements of Out(F r ). Diretions nd turns In generl we use definitions from [BH9] nd [BFH00] for disussing trin trk mps. We remind the reder of dditionl definitions nd nottion given in [Pf1]. Here Γ will e rose nd g : Γ Γ will represent φ Out(F r ). E + (Γ) := {E 1,..., E n } = {e 1, e 1,..., e n 1, e n } will e the edge set of Γ with presried orienttion. E(Γ):= {E 1, E 1,..., E n, E n }, where E i denotes E i oppositely oriented. If n edge indexing {E 1,..., E n } (thus indexing {e 1, e 1,..., e n 1, e n }) is presried, Γ is lled edge-indexed. D(v) or D(Γ) will denote the set of diretions t the vertex v. For eh e E(Γ), D 0 (e) will denote the initil diretion of e. Also, D 0 γ := D 0 (e 1 ) for ny pth γ = e 1... e k in Γ. Dg will denote the diretion mp g indues. We ll d D(Γ) periodi if Dg k (d) = d for some k > 0 nd fixed if k = 1. T (v) will denote the set of turns t v nd D t g the indued turn mp. Sometimes we usively write {e i, e j } for {D 0 (e i ), D 0 (e j )}. For pth γ = e 1 e... e k 1 e k in Γ, we sy γ trverses {e i, e i+1 } for eh 1 i < k. Rell tht turn is lled illegl for g if Dg k (d i ) = Dg k (d j ) for some k. Trnsition mtries nd irreduiility. The trnsition mtrix for topologil representtive g is the squre mtrix where the ij th entry is the numer of times g(e j ) trverses E i in either diretion. A mtrix A = [ ij ] is irreduile if eh entry ij 0 nd if, for eh i nd j, there exists k > 0 so tht the ij th entry of A k is stritly positive. The mtrix is Perron-Froenius (PF) if eh suffiiently high k works for ll index pirs {i, j}. [BH9] The Full Irreduiility Criterion will require tht the trnsition mtrix for representtive e PF. It will e relevnt tht ny power of PF mtrix is oth PF nd irreduile. Additionlly, topologil representtive is irreduile if nd only if its trnsition mtrix is irreduile [BH9]. Periodi Nielsen pths nd geometri outer utomorphisms. Rell [BH9], for g : Γ Γ, nontrivil pth ρ in Γ is lled periodi Nielsen pth (pnp) if, for some k, g k (ρ) ρ rel endpoints. For k = 1, ρ is lled Nielsen pth (Np). ρ is lled n indivisile Nielsen pth (inp) if it nnot e written s nontrivil ontention ρ = ρ 1 ρ of Np s ρ 1 nd ρ. If ρ is n inp for n irreduile trin trk mp g, then (Lemm.4, [BF94]) there exist unique, nontrivil, legl pths α, β, nd τ in Γ so tht ρ = ᾱβ, g(α) = τα, nd g(β) = τβ. In [BF94], immersed pths α 1,..., α k Γ re sid to form n orit of periodi Nielsen pths if g k (α i ) α i+1 mod k rel endpoints, for ll 1 i k. The orit is lled indivisile when α 1 is not ontention of supths elonging to orits of pnps. Eh α i in n indivisile orit is lled n indivisile periodi Nielsen pth (ipnp). Rell [GJLL98], tht n outer utomorphism is geometri whose stle representtive, in the sense of [BH9], hs no pnp s. We denote y AFI r the suset of FI r onsisting of the geometri elements. Idel Whitehed grphs nd lmintion trin trk (ltt) strutures. We remind the reder of n [HM11] idel Whitehed grph definition nd [Pf1] lmintion trin trk struture definition. See [Pf1] nd [HM11] for desriptions of idel Whitehed grph lterntive definitions nd outer utomorphism invrine. Note tht, while we use representtive to onstrut it, the idel Whitehed grph does not depend on the hoie of representtive for n outer utomorphism. Let Γ e mrked grph, g : Γ Γ tt representive of φ Out(F r ), nd v Γ singulrity (the endpoint of n ipnp or vertex with t lest three periodi diretions). The lol Whitehed grph LW(g; v) for g t v hs: (1) vertex for eh diretion d D(v) nd () edges onneting verties for d 1, d D(v) when {d 1, d } is tken y some g k (e), with e E(Γ). The lol stle Whitehed grph SW(g; v) is the sugrph otined y restriting preisely to verties

4 with periodi diretion lels. For rose Γ with vertex v, we denote the single lol stle Whitehed grph SW(g; v) y SW(g) nd the single lol Whitehed grph LW(g; v) y LW(g). For pnp-free g, the idel Whitehed grph IW(φ) of φ is isomorphi to SW(g; v). In singulrities v Γ prtiulr, when Γ is rose, IW(φ) = SW(g). Let g e pnp-free tt mp on mrked rose Γ with vertex v. Rell from [Pf1] the definition of the lmintion trin trk (ltt) struture G(g) for g: The olored lol Whitehed grph CW(g) t v is LW(g) with the sugrph SW(g) olored purple nd LW(g) SW(g) red (nonperiodi diretion verties re red). Let Γ N = Γ N(v), where N(v) is ontrtile neighorhood of v. For eh E i E +, dd verties D 0 (E i ) nd D 0 (E i ) t the orresponding oundry points of the prtil edge E i (N(v) E i ). G(g) is formed from Γ N CW(g) y identifying the vertex di in Γ N with the vertex d i in CW(g). Nonperiodi diretions verties re red, edges of Γ N re lk, nd periodi verties re purple. G(g) is given smooth struture vi prtition of the edges into the set of lk edges E nd the set of olored edges E. A smooth pth will men pth lternting etween olored nd lk edges. We refer the reder to [Pf1] or [Pf1] for thorough presenttion of strt lmintion trin trk strutures. We summrize just severl definitions here. Rell tht trin trk (tt) grph is finite grph G stisfying: tt1: G hs no vlene-1 verties; tt: eh edge of G hs distint verties (single edges re never loops); nd tt: the edge set of G is prtitioned into two susets, E (the lk edges) nd E (the olored edges), suh tht eh vertex is inident to t lest one E E nd t lest one E E. A lmintion trin trk (ltt) struture G is pir-leled olored trin trk grph (lk edges will e inluded, ut not onsidered olored) stisfying: ltt1: Verties re either purple or red. ltt: Edges re of types (E omprises the lk edges nd E omprises the red nd purple edges): (Blk Edges): A single lk edge onnets eh pir of (edge-pir)-leled verties. There re no other lk edges. In prtiulr, eh vertex is ontined in unique lk edge. (Red Edges): A olored edge is red if nd only if t lest one of its endpoint verties is red. (Purple Edges): A olored edge is purple if nd only if oth endpoint verties re purple. ltt: No pir of verties is onneted y two distint olored edges. We denote the purple sugrph of G (from SW(g)) y PI(G) nd, if G = PI(G), sy G is n ltt struture for G. An (r; ( r)) ltt struture is n ltt struture G for G PI (r;( r)) suh tht: ltt(*)4: G hs preisely r-1 purple verties, unique red vertex, nd unique red edge. We onsider ltt strutures equivlent tht differ y n ornmenttion-preserving homeomorphism nd refer the reder to the Stndrd Nottion nd Terminology. of [Pf1]. In prtiulr, in strt nd nonstrt ltt strutures, [d i, d j ] is the edge onneting vertex pir {d i, d j }, [e i ] denotes the lk edge [d i, d i ] for e i E(Γ), nd C(G) denotes the olored sugrph (from LW(g)). Purple verties re periodi nd red verties nonperiodi. G is dmissile if ireurrent s trin trk struture (i.e hs lolly smoothly emedded line trversing eh edge infinitely mny times s R nd s R ). For n (r; ( r)) ltt struture G for G, dditionlly: 1. d u lels the unique red vertex nd is lled the unhieved diretion.. e R = [t R ] denotes the unique red edge, d lels its purple vertex, thus t R = {d u, d } (e R = [d u, d ]).. d is ontined in unique lk edge, whih we ll the twie-hieved edge. 4. d will lel the other twie-hieved edge vertex nd e lled the twie-hieved diretion. 5. If G hs susript, the susript rries over to ll relevnt nottion. For exmple, in G k, d u k will lel the red vertex nd e R k the red edge. 4

5 We ll r-element set of the form {x 1, x 1,..., x r, x r }, with elements pired into edge pirs {x i, x i }, rnk-r edge pir leling set (we write x i = x i ). We ll grph with verties leled y n edge pir leling set pir-leled grph, nd n indexed pir-leled grph if n indexing is presried. A G PI (r;( r)) is (index) pir-leled whose verties re leled y r 1 element suset of the rnk-r (indexed) edge pir leling set. An ltt struture, index pir-leled s grph, is n indexed pir-leled ltt struture if the verties of the lk edges re indexed y edge pirs. Index pir-leled ltt strutures re equivlent tht re equivlent s ltt strutures vi n equivlene preserving the indexing of the vertex leling set. By rnk-r index pir-leling n (r; ( r)) ltt struture G nd edge-indexing the edges of n r-petled rose Γ, one retes n identifition of the verties in G with D(v), where v is the vertex of Γ. With this identifition, we sy G is sed t Γ. In suh se we my use the nottion {d 1, d,..., d r 1, d r } for the vertex lels. Additionlly, [e i ] denotes [D 0 (e i ), D 0 (e i )] = [d i, d i ] for eh edge e i E(Γ). We ll permuttion of the indies 1 i r omined with permuttion of the elements of eh pir {x i, x i } n edge pir (EP) permuttion. Edge-indexed grphs will e onsidered edge pir permuttion (EPP) isomorphi if there is n EP permuttion mking the lelings identil. Idel deompositions. M. Feighn nd M. Hndel defined rottionless trin trk representtives nd outer utomorphisms in [FH11]. Rell [HM11]: Let φ AFI r e suh tht IW(φ) PI (r;( r)), then φ is rottionless if nd only if the verties of IW(φ) PI (r;( r)) re fixed y the tion of φ. The following is Proposition. of [Pf1]: Proposition. [Pf1] Let φ AFI r with IW(φ) PI (r;( r)). There exists pnp-free tt mp on the rose representing rottionless power ψ = φ R g 1 g g n 1 g n nd deomposing s Γ 0 Γ1 Γ n 1 Γn, suh tht: (I) the index set {1,..., n} is viewed s the set Z/nZ with its nturl yli ordering; (II) eh Γ k is n edge-indexed rose with n indexing {e (k,1), e (k,),..., e (k,r 1), e (k,r) } where: () one n edge-index Γ with E(Γ) = {e 1, e,..., e r 1, e r } suh tht, for eh t with 1 t r, g(e t ) = e i1... e is where (g n g 1 )(e 0,t ) = e n,i1... e n,is ; () for some i k, j k with e k,ik (e k,jk ) ±1, g k (e k 1,t ) := { e k,t e k,jk for t = i k e k,t for ll e k 1,t e ±1 k 1,j k ; nd () for eh e t E(Γ) suh tht t j n, we hve Dh(d t ) = d t, where d t = D 0 (e t ). Rell tht tt mps stisfying (I)-(II) re lled idelly deomposle (ID) with n idel deomposition (ID). An ID g suh tht φ AFI r nd IW(φ) PI (r;( r)) hs type (r; ( r)). In [Pf1] we proved for (r; ( r)) tt mp g : Γ Γ, tht G(g) is n (r; ( r)) ltt struture with se Γ. Agin we denote e k 1,jk y e pu k 1, denote e k,j k y e u k, denote e k,i k y e k, nd denote e k 1,i k 1 y e p k 1. Also,D k := D(Γ k ), E k := E(Γ k ), nd G k := G(f k ) where f k := g k g 1 g n g k+1 : Γ k Γ k. And { g k g i : Γ i 1 Γ k if k > i nd g k,i :=. g k g 1 g n g i if k < i It is proved in [Pf1] tht D 0 (e u k ) = du k, D 0(e k ) = d k, D 0(e pu k 1 ) = dpu k 1, nd D 0(e p k 1 ) = dp k 1. As desried in [Pf1], for ny k, l, we hve diretion mp Dg k,l, n indued mp of turns Dgk,l t, nd n indued mp of ltt strutures Dgk,l T : G l 1 G k. Dgk,l C denotes the restrition to C(G l 1) of Dgk,l T. 5

6 Extensions nd swithes. By proper full fold we men the identifition of (proper) prtil edge with full edge. A triple (g k, G k 1, G k ) is n ordered set of three ojets where g k : Γ k 1 Γ k is proper full fold of roses nd, for i = k 1, k, G i is n ltt struture with se Γ i. Rell [Pf1], in n ID of (r; ( r)) representtive, eh (g k, G k 1, G k ) stisfies the dmissile mp properties AMI-VII of [Pf1] nd is either swith or extension. A generting triple (gt) is triple (g k, G k 1, G k ) where (gti) g k : Γ k 1 Γ k is proper full fold of edge-indexed roses defined y. g k (e k 1,jk ) = e k,ik e k,jk where d k = D 0(e k,ik ), d u k = D 0(e k,jk ), nd e k,ik (e k,jk ) ±1 nd. g k (e k 1,t ) = e k,t for ll e k 1,t (e k,jk ) ±1 ; (gtii) G i is n indexed pir-leled (r; ( r)) ltt struture with se Γ i for i = k 1, k; nd (gtiii) The indued mp of sed ltt strutures D T (g k ) : G k 1 G k exists nd, in prtiulr, restrits to n isomorphism from PI(G k 1 ) to PI(G k ). G k 1 is the soure ltt struture nd G k the destintion ltt struture. If oth re dmissile, the triple is dmissile. We sometimes write g k : e pu k 1 e k eu k for g k, write d pu k 1 for d k 1,j k, nd write e p k 1 for e k 1,ik. If G k nd G k 1 re index pir-leled (r; ( r)) ltt strutures for G, then (g k, G k 1, G k ) will e generting triple for G. The swith determined y purple edge [d k, d (k,l)] in G k is the gt (g k, G k 1, G k ) for G stisfying: (swi): D T (g k ) restrits to n isomorphism from PI(G k 1 ) to PI(G k ) defined y d pu k 1 d k = d k,i k (d k 1,t d k,t for d k 1,t d pu k 1 ) nd extended linerly over edges. (swii): d p k 1 = du k 1. (swiii): d k 1 = d k 1,l. pu d k-1 =d k-1,jk G k-1 g k e e e k-1,j k k,i k k,j k d u k = dk,j k e u k G k u d k-1 e pu k-1 p = d k-1,ik =d k-1 d k-1,l d k-1 e k-1 e pu k-1 e e k u k e k d k= dk,i k d k,l The extension determined y [d k, d k,l], is the gt (g k, G k 1, G k ) for G stisfying: (exti): The restrition of D T (g k ) to PI(G k 1 ) is defined y sending, for eh j, the vertex leled d k 1,j to the vertex leled d k,j nd extending linerly over edges. (extii): d u k 1 = dpu k 1, i.e. dpu k 1 = d k 1,j k lels the single red vertex in G k 1. (extiii): d k 1 = d k 1,l. u G =dk-1,jk g k-1 e k e e d k-1 pu = d k-1 e pu k-1 e r k-1 d k-1 d k-1,l k-1,j k e pu k-1 k,i k e e k u k k,j k e d u k = d k k,j k e k r e u k d = k dk,i k G k d k,l 6

7 Compositions of extensions nd swithes Compositions of sequene of extensions or sequene of swithes ply n importnt role in our proofs. Definition.1. A predmissile omposition (g i k,..., g i, G i k 1,..., G i ) for G PI (r;( r)) is sequene of proper full folds of (edge-pir)-indexed roses, Γ i k 1 g i k Γ i k gi 1 g i Γ i 1 Γi, with ssoited sequene of (r; ( r)) ltt strutures for G, G i k 1 D T (g i k ) G i k D T (g i k+1 ) DT (g i 1 ) G i 1 D T (g i ) G i, where, for eh i k 1 j < i, (g j+1, G j, G j+1 ) is n extension or swith for G. The Definition.1 nottion will e stndrd. A omposition is dmissile if eh G j is. We ll g i,i k the ssoited utomorphism, G i k 1 the soure ltt struture, nd G k the destintion ltt struture. To ensure IW(g) = C r in Theorem 6., we use uilding lok ompositions of extensions: If eh (g j, G j 1, G j ) with i k < j i is n dmissile extension nd (g i k, G i k 1, G i k ) is n dmissile swith, then we ll (g i k,..., g i ; G i k 1,..., G i ) n dmissile onstrution omposition for G. We ll g i,i k onstrution utomorphism. Leving out the swith, gives purified onstrution utomorphism g p = g i g i k+1 nd purified onstrution omposition (g i k+1,..., g i ; G i k,..., G i ). A onstrution utomorphism lwys hs the form of Dehn twist utomorphism e pu i k 1 weu i k, where w = e i k... e i. One n view the omposition s twisting the edge orresponding to epu i k 1 round the pth orresponding to w in the destintion ltt struture. In the next setion we desrie these pths nd prove (Proposition.6) they onstrut smooth pth in their destintion ltt struture..1 Constrution Pths Corresponding to onstrution omposition is pth in its destintion ltt struture. A key property of suh pth (Lemm.6) holds when the onstrution omposition is prt of the idel deomposition of type (r; ( r)) representtive g: the imge of the pth s purple edges live in G(g). We use nottion throughout this setion y dropping indies. While not neessry, it my id in visuliztion of the properties nd proedures, s well s redue potentil onfusion over indies. Lemm.. Let (g 1,..., g n, G 0,..., G n ) e n ID for G PI (r;( r)) nd (g i k,..., g i ; G i k 1,..., G i ) onstrution omposition. Then [d u i, d i, d i, d i, d i 1, d i 1,..., d i k+1, d i k+1, d i k ] = [du i, d i, d i, d i 1,..., d i k, d i k ] is smooth pth in the ltt struture G i. Proof. We proeed y indution for deresing s. Proof y indution is vlid, s the proof does not rely on G i k 1 (the only thing distinguishing (g i k, G i k 1, G i k ) s swith). For the se se note tht e R i = [d u i, d i ]. So [du i, d i, d i ] is pth in G i nd smooth, s it lterntes etween olored nd lk edges ([d u i, d i ] is olored nd [d i, d i ] is lk). For the indution ssume, for i > s > i k, [d u i, d i, d i, d i 1, d i 1,..., d s+1, d s, d s] is smooth pth in G i (ending with the lk edge [d s, d s]). By [Pf1] Corollry 5.6, e R s 1 = [du s 1, d s 1 ]. By [Pf1] Lemm 5.7, DC g s ([d u s 1, d s 1 ]) = [d s, d s 1 ] is purple edge in G s. Sine purple edges re lwys mpped to themselves y extensions (in the sense tht D C preserves the seond index of their vertex lels) nd D C g s ([d u s, d s 1 ]) = [d s, d s 1 ] is in PI(G s ), D C g n,s ({d u s 1, d s 1 }) = DC g n,s+1 (D C g s ([d u s 1, d s 1 ])) = DC g n,s+1 ([d s, d s 1 ]) = [d s, d s 1 ] is in 7

8 PI(G i ). Thus, inluding the purple edge [d s, d s 1 ] in the smooth pth [du i, d i, d i, d i 1, d i 1,..., d s+1, d s, d s] gives the smooth pth [d u i, d i, d i, d i 1,..., d s+1, d s, d s, d s 1 ]. (It is smooth, s we dded olored edge to pth with edges lternting etween olored nd lk, ending with lk). By inluding the lk edge [d s 1, d s 1 ] we get the onstrution pth [du i, d i, d i, d i 1, d i 1,..., d s, d s 1, d s 1 ]. (Also smooth, s we dded lk edge to pth with edges lternting etween olored nd lk, ending olored). This onludes the indutive step, hene proof. The pth of Lemm. (depited in Exmple.5) is lled the onstrution pth for (g i k,..., g i ; G i k 1,..., G i ) nd denoted γ gi,i k. One otins it y trversing the red edge [d u i, d i ] from the red vertex d u i to the vertex d i, the lk edge [d i, d i ] from d i to d i, the extension determining purple edge [d i, d i] = [d i, d i 1 ] from d i to d i = d i 1, the lk edge [d i 1, d i 1 ] from d i 1 to d i 1, the extension determining purple edge [d i 1, d i 1] = [d i 1, d i ] from d i 1 to d i 1 = d i, the lk edge [d i, d i ] from d i to d i, ontinuing s suh through the purple edges determining eh g j (inserting lk edges etween), nd finlly trversing [d i k+1, d i k+1] = [d i k+1, d i k ] nd then [d i k, d i k ] from d i k to d i k. Let G e n dmissile (r; ( r)) ltt struture with the stndrd nottion. The onstrution sugrph G C is onstruted from G vi the following proedure: 1. Remove the interior of the lk edge [e u ], the purple vertex d u, nd the interior of ny purple edges ontining the vertex d u. Cll the grph with these edges nd verties removed G 1.. Given G j 1, reursively define G j : Let {α j 1,i } e the set of verties in G j 1 not ontined in ny olored edge of G j 1. G j is otined from G j 1 y removing ll lk edges ontining vertex α j 1,i {α j 1,i }, s well s the interior of eh purple edge ontining vertex α j 1,i.. G C = j G j. A onstrution pth tully lwys lives in the onstrution sugrph of its destintion ltt struture. Exmple.. To find the onstrution sugrph G C for the ltt struture G on the left (1), we remove the interior of the lk edge [ā, ] to otin the middle grph (), then remove nd the interior of ll purple edges ontining to otin G C (grph () depited on the right) The following lemm gives some onditions under whih pth in n dmissile (edge-pir)-indexed (r; ( r)) ltt struture G is gurnteed to e the onstrution pth for onstrution omposition with destintion ltt struture G. It lso explins how to find suh onstrution omposition. Lemm.4. Let G e n dmissile (r; ( r)) ltt struture nd onsider smooth pth γ = [d u, x 1, x 1, x, x,..., x k+1, x k+1 ] in G C strting with e R (oriented from d u to d ) nd ending with the lk edge [x k+1, x k+1 ]. Edge-index r-petled roses Γ i k 1,..., Γ i nd define the homotopy equivlenes Γ i k 1 g i k Γ i k g i k+1 gi 1 Γ i 1 g i Γi y g l : e l 1,s e l,tl e l,s, where D 0 (e l,tl ) = x i l+1, nd g l (e l 1,j ) = e l,j for e l 1,j e ±1 l 1,s. Define the ltt strutures (with respetive ses Γ j ) G t, for i k 1 t i, y hving: 1. eh PI(G l ) isomorphi to PI(G i ) vi n isomorphism preserving the vertex lel seond indies, 8

9 . the seond index of the lel on the single red vertex in eh G l e s (the sme s in G i ), nd. the single red edge in G l e [d l,s, d l,tl ]. If eh G j is n dmissile (r; ( r)) ltt struture for G with se Γ j, then (g i k,..., g i ; G i k 1,..., G i ) is purified onstrution omposition with onstrution pth γ. For eh i k + 1 l i, the triple (g l, G l 1, G l ) is the extension determined y [x i l+1, x i l+ ]. Proof. It suffies to show: A. eh (g l, G l 1, G l ) is the extension determined y [x i l+1, x i l+ ] (so tht (g i k,..., g i ; G i k 1,..., G i ) is indeed onstrution omposition) nd B. the orresponding onstrution pth is [d u i, d i, d i, d i 1, d i 1,..., d i k+1, d i k, d i k ]. (exti) holds y our requiring eh G j e n (r; ( r)) ltt struture with rose se grph. The G l re (r; ( r)) ltt strutures for PI(G) y (1)-() in the lemm sttement. This, with how we defined our nottion, implies (gtiii) nd (exti). The seond index of the red vertex lel is the sme in eh G l s in G i, giving (ext II). To see (extiii) holds y (1), note tht [x i l+1, x i l+ ] is in PI(G l ) (it is in G nd PI(G) = PI(G l )) nd would e the determining edge for the extension. (A) is proved. The onstrution pth is [d u i, d i, d i, d i 1, d i 1,..., d i k+1, d i k, d i k ] y Lemm., proving (B). It is proved in [Pf1] tht (g i k,..., g i ; G i k 1,..., G i ) is in ft the unique onstrution omposition with γ s its onstrution pth. We ll Γ i k 1 Γ i k Γ i 1 Γi, together g i k g i k+1 g i 1 g i D with its sequene of ltt strutures G T (g i k ) D i k 1 G T (g i k+1 ) i k DT (g i 1 ) D G T (g i ) i 1 G i, s in the lemm, the onstrution omposition determined y the pth γ = [d u, x 1, x 1, x, x,..., x k+1, x k+1 ]. Exmple.5. In the following ltt struture, G i, the numered edges give onstrution pth determined y the onstrution utomorphism (the utomorphism fixes ll other edges) We retrieve eh ltt struture G i k in the onstrution omposition y moving the red edge of G i to e tthed to the terminl vertex of edge k in the onstrution pth. If the red vertex of G j is d s nd the red edge is [d s, d t ], then g j is defined y e s ē t e s. We show the onstrution omposition, leving out the soure ltt struture G i 7 of the swith to highlight tht it does not ffet the onstrution pth. i-6 i-5 i-4 g g g g i -6 i -5 i -4 i - G i- G G G G 5 i- i-1 i g g g i- i -1 i Lemm.6 is fundmentl to our onstrution tehniques. It sys tht onstrution ompositions, in ft, uild in the idel Whitehed grph the imges of the purple edges of the onstrution pth: 9 G G

10 Lemm.6. Let g e n ID type (r; ( r)) representtive of φ Out(F r) with IW(φ) = G. Suppose g 1 g g n 1 g n g deomposes s Γ = Γ 0 Γ1 Γ n 1 Γn = Γ, with the sequene of ltt strutures for G: G i k 1 D T (g i k ) G i k D T (g i k+1 ) DT (g i 1 ) G i 1 D T (g i ) G i. If g = g n g k+1 is onstrution omposition, then G ontins s sugrph the purple edges in the onstrution pth for g. Proof. We proeed y indution for deresing k. Proof y indution is vlid here sine nothing in the proof will rely on G k (the only thing distinguishing (g k, G k, G k+1 ) s swith insted of n extension). For the se se onsider g n g n 1. By [Pf1] Corollry 5.6 G n 1 hs red edge [d u n 1, d n 1 ]. We know g n is defined y g n : e pu n 1 e ne u n nd g n (e n 1,l ) = e n,l for ll e n 1,l (e pu n 1 )±1. Thus, sine d pu n 1 = du n 1 d n 1, we know tht Dg n(d n 1 ) = d n 1. So DC g n ([d u n 1, d n 1 ]) = DC g n ([d pu n 1, d n 1 ]) = [d n, d n 1 ] nd, sine DC g n (C(G n 1 )) PI(G n ), [d n, d n 1 ] is in PI(G n). The se se is proved. For the indutive step ssume, for n > s > k + 1, G n ontins the purple edges of γ gn,s. Agin y [Pf1] Corollry 5.6, e R s 1 = [du s 1, d s 1 ]. As ove, DC g s ([d u s 1, d s 1 ]) = [d s, d s 1 ] is in PI(G s). Sine extensions mp purple edges to themselves nd D C g s ([d u s, d s 1 ]) = [d s, d s 1 ], DC g n,s ([d u s 1, d s 1 ]) = D C g n,s+1 (D t g s ([d u s 1, d s 1 ])) = DC g n,s+1 ([d s, d s 1 ]) = [d s, d s 1 ], proving the indutive step.. Swith Pths We use swith pths to find swith sequenes. Here swith sequenes ply two primry roles: ensuring our idel deomposition tully gives loop in ID(G) nd ensuring our trnsition mtrix is PF. We ontinue with the nottionl use of the previous setion (primrily ignoring seond indies). Definition.7. (See Exmple.1) An dmissile swith sequene for (r; ( r)) grph G is n dmissile omposition (g i k,..., g i ; G i k 1,..., G i ) for G suh tht (ss1) eh (g j, G j 1, G j ) with i k j i is swith nd (ss) d n+1 = du n d u l = d l+1 nd d l d u n = d n+1 for ll i n > l i k. We ll the ssoited utomorphism g i,i k = g i g i k swith sequene utomorphism. Remrk.8. (ss) is not implied y (ss1) nd is neessry for swith pth to indeed e pth. Certin sttements in the Lemm.11 proof elow (showing tht the swith pth for swith sequene is smooth pth in the destintion ltt struture) would e inorret without (ss). Definition.9. Let (g j,..., g k ; G j 1,..., G k ) e n dmissile swith sequene. Its swith pth is pth in the destintion ltt struture G k trversing the red edge [d u k, d k ] from its red vertex du k to d k, the lk edge [d k, d k ] from d k to d k, wht is the red edge [du k 1, d k 1 ] = [d k, d k 1 ] in G k 1 (purple edge in G k ) from d k = du k 1 to d k 1, the lk edge [d k 1, d k 1 ] from d k 1 to d k 1, ontinues s suh through the red edges for the G i with j i k (inserting lk edges etween), nd ends y trversing the lk edge [d j+1, d j+1 ] from d j+1 to d j+1, wht is the red edge [du j, d j ] = [d j+1, d j ] in G j (purple edge in G k ), nd then the lk edge [d j, d j ] from d j to d j. In other words, swith pth lterntes etween the red edges (oriented from d u j to d j ) for the G j (for desending j) nd the lk edges etween. Remrk.10. We lrify here some wys in whih swith pths nd onstrution pths differ: 1. The purple edges in the onstrution pth for onstrution omposition (g i k,..., g i ; G i k 1,..., G i ) re purple in eh G l with i l l < i, for swith pth. They re red edges in the struture G l they 10

11 re reted in nd then will not exist t ll in the strutures G m with m < l. The hnge of olor (nd dispperne) of red edges is the reson for (ss) in the swith sequene definition.. Unlike onstrutions pths, swith pths do not give supths of lmintion leves. The following lemm proves tht swith pths re indeed smooth pths in destintion LTT strutures. It is importnt to note tht this only holds when (ss1) nd (ss) hold. Lemm.11. Let (g 1,..., g n, G 0,..., G n ) e n ID for G PI (r;( r)) nd (g i k,..., g i ; G i k 1,..., G i ) swith sequene. Then the ssoited swith pth forms smooth pth in the ltt struture G k. Proof. The red edge in G k is [d u k, d k ]. We re left to show (y indution) tht: (1) For eh 1 l < k, [d u l, d l ] = [d l+1, d l ] is purple edge of G k nd () lternting the purple edges [d l+1, d l ] with the lk edges [d l, d l ] gives smooth pth in G k. We prove the se se. By the swith properties, e R k 1 is [du k 1, d k 1 ] = [d k, d k 1 ]. Sine d k du k nd d k 1 du k (y the swith sequene definition), Dt g k ({d k, d k 1 }) = {d k, d k 1 }. So [d k, d k 1 ], is purple edge in G k. Sine e R k = [du k, d k ], y inluding the lk edge [d k, d k ], we hve pth [du k, d k, d k, d k 1 ] in G k (smooth, s it lterntes etween olored nd lk edges). The se se is proved. We prove the indutive step. By the indutive hypothesis we ssume the sequene of swithes for g k,..., g k i gives us smooth pth [d u k,..., d k i ] in G k ending with purple edge with free vertex d k i 1. We know er k i 1 = [du k i 1, d k i 1 ] = [d k i, d k i 1 ]. As long s du l d k i nd du l d k i 1 for k i l k (holding y the swith sequene definition), Dt g k,k i ({d u (k i 1), d (k i 1) }) = D t g k,k i ({d (k i), d (k i 1) }) = {d (k i), d (k i 1) }. So, [d (k i), d (k i 1) ] is purple edge in G k. Sine [d u k,..., d k i ] is smooth pth in G k ending with lk edge, [d u k,..., d k i, d k i, d k i 1 ] is lso smooth pth in G k, s [d k i, d k i ] is lk edge in G k nd [d k i, d k i 1 ] purple edge in G k. Exmple.1. In the ltt struture G i of Exmple.5 we numer the olored edges of swith pth: 0 The swith sequene onstruted from the swith pth is: Gk- Gk G -1 k g g k -1 k The red edge e r k in G k is (0), the red edge e r k 1 in G k 1 is (1), nd the red edge e r k in G k is (). The following lemm explins how, y inserting onstrution ompositions into well-hosen swith sequene, one n ensure their trnsition mtrix is PF (for purposes of pplying the FIC). Lemm.1. Suppose g deomposes s g i m,i m g i 1,i 1 where: 1. Eh g i k,i k is onstrution omposition whose pure onstrution omposition strts nd ends with the sme ltt struture.. For eh edge pir {d i, d i }, either d i or d i is the red unhieved diretion vertex for some G ik

12 Then the trnsition mtrix for g is PF. Proof. It suffies to show eh e u i k is in the imge of eh e u i j. In ft, it suffies to show eh e u i k 1 is in the imge of eh e u i k. Note g ik (e pu (i k 1) ) = e i k e u i k. Sine (g ik, G (ik 1), G ik ) is swith, e p (i k 1) = eu (i k 1). Sine g i (k 1),i (k 1) is onstrution omposition whose pure onstrution omposition strts nd ends with the sme ltt struture, e u (i k 1) = eu i (k 1). So e p (i k 1) = eu i (k 1) nd e u i k mps over e u i k 1. (e u i k hs the sme seond index s e pu i k nd e u i k 1 hs the sme seond index s e i k. Use Lemm 5. of [Pf1].) 4 Full Irreduiility Criterion We prove Folk Lemm giving riterion, the Full Irreduiility Criterion (FIC), for n irreduile tt representtive to e fully irreduile. Our originl pproh involved the Wek Attrtion Theorem, severl notions of trin trks, lmintions, nd the sin of ttrtion for lmintion. However, Mihel Hndel griously provided method to omplete it, mking muh of our initil work unneessry. Our proof here uses Hndel s reommendtion. [K1] gives riterion sed on ours. The proof we give uses the reltive trin trks (rtts) of [BH9] nd the ompletely split reltive trin trks (CTs) of [FH11]. If φ Out(F r ) is rottionless nd C is nested sequene of φ-invrint free ftor systems, then φ is represented y CT nd filtrtion relizing C ([FH11], Theorem 4.9). We use from the CT definition (CT5): for fixed strtum H t with unique edge E t, either E t is loop or eh end of E t is in Γ t 1. Our definition of the ttrting lmintion for n outer utomorphism will e s in [BFH00]. A omplete summry of relevnt definitions n e found in [Pf1]. L(φ) will denote the set of ttrting lmintions for φ. By [BFH00], for φ Out(F r ), there exists orrespondene etween L(φ) nd the EG-strt of rtt representtive g : Γ Γ of φ: For eh EG strtum H t, there exists unique ttrting lmintion Λ t with H t s the highest strtum rossed y the reliztion λ Γ of Λ t -generi line. Λ(φ) will denote the unique ttrting lmintion for n irreduile φ. Free ftor support is defined in [BFH00] (Corollry.6.5). The relevnt informtion for our FIC proof is: if lmintion is rried y proper free ftor, then its support is proper free ftor. Proposition 4.1. (The Full Irreduiility Criterion) Let g e trin trk representive of n outer utomorphism φ Out(F r ) suh tht (I) g hs no periodi Nielsen pths, (II) the trnsition mtrix for g is Perron-Froenius, nd (III) ll lol Whitehed grphs LW (x; g) for g re onneted. Then φ is fully irreduile. Proof. Suppose g : Γ Γ is s in the sttement. Sine g hs PF trnsition mtrix, s n rtt, it hs preisely one strtum nd tht strtum is EG. Hene, it hs preisely one ttrting lmintion [BFH00]. Sine the numer of ttrting lmintions for tt representtive is independent of the representtive hoie, ny φ representtive would lso hve preisely one ttrting lmintion. Suppose, for ontrdition s ske, φ were not fully irreduile. Then some φ k would e reduile. If neessry, tke n even higher power R so tht φ R is lso rottionless (this does not hnge reduiility). Note, sine L(φ) is φ-invrint, ny φ R representtive lso hs preisely one ttrting lmintion. Sine φ R is reduile (nd rottionless), there exists CT representtive h : Γ Γ of φ R with more thn one strtum ([FH11], Theorem 4.9). Sine φ R hs preisely one ttrting lmintion, h hs preisely one EG-strtum H t. Eh strtum H i, other thn H t nd ny zero strt, would e n 1

13 NEG-strtum onsisting of single edge E i ([FH11], Lemm 4.). We onsider seprtely the ses where t = 1 nd t > 1. Sine the trnsition sumtrix for ny zero strtum is zero (hene every edge of the strtum is mpped to lower filtrtion element y h), H 1 ould not e zero strtum. Thus, if t > 1, then H 1 is NEG nd must onsist of single edge E 1. Sine H 1 is ottom-most, it would e fixed, s there re no lower strt for its edge to e mpped into. Aording to (CT5), E 1 would then e n invrint loop. This would imply φ R hs rnk-1 invrint free ftor. However, g (hene g R ) ws pnp-free. So φ R ould not hve rnk-1 invrint free ftor. We hve rehed ontrdition for t > 1. Assume t = 1. Then Λ(φ R )(= Λ(φ)) is rried y proper free ftor. Proposition.4 of [BFH97] sys, if finitely generted sugroup A F r rries Λ φ, then A hs finite index in F r. The neessry onditions for this proposition re tully only: 1. the trnsition mtrix of g is irreduile nd. eh LW (g; v) is onneted. (Up to the ontrdition in the proof of Proposition.4 of [BFH97], the only properties used in the proof re tht the support is finitely generted, proper, nd rries the lmintion. The ontrdition uses Lemm.1 in [BFH97], whih shows (1) nd () rry over to lifts of g to finitesheeted overing spes, using no properties other thn properties (1) nd ().) Assumptions (1) nd () re ssumptions in this lemm s hypotheses nd Λ(φ) is still the ttrting lmintion for g. So we n pply the proposition to ontrdit Λ(φ) hving proper free ftor support: Applying the proposition, sine proper free ftors hve infinite index, the support must e the whole group. This ontrdits tht the EG-strtum is H 1 nd tht there must e more thn one strtum. We hve thus shown tht we nnot hve more thn one strtum with t = 1 or t > 1. So ll powers of φ must e irreduile nd φ is fully irreduile, s desired. Lemm 4.. Suppose G PI (r;( r)) nd (g 1,..., g k ; G 0,..., G k ), g = g k g 1, is n ID, stisfying: 1. PI(G(g)) = G. More preisely, G = SW(g; v).. And for eh 1 i, j q, there exists some k 1 suh tht g k (E j ) ontins either E i or Ēi.. And g hs no periodi Nielsen pths. Then g is tt representtive of φ AFI r suh tht IW(φ) = G. Proof. By the FIC, we only need to show g is tt mp, the trnsition mtrix of g is PF, nd IW(φ) = G. () implies tht the trnsition mtrix is PF. g is tt mp y [Pf1] Lemm 5.. Sine g hs no pnp s, IW(g) = SW(g). By the definition of G(g), we know PI(G(g)) = SW(v; g). Remrk 4.. In the lnguge of [Pf1], the onditions of Lemm 4. ould e stted s L(g 1,..., g k ; G 0, G 1..., G k 1, G k ) = E(g 1, G 0, G 1 ) E(g k, G k 1, G k ) eing loop in ID(G) stisfying (1)-() nd induing mp fixing ll periodi diretions. 5 Nielsen pth identifition nd prevention We give (Proposition 5.) method for finding ll ipnp s, thus pnp s, for tt mp g : Γ Γ idelly g 1 g g n 1 g n deomposed s Γ = Γ 0 Γ1 Γ n 1 Γn = Γ where (g 0,..., g n ; G 0,..., G n ) is n dmissile omposition. Note tht, for eh k, we know tht T k = {d pu k, dp k } is the unique illegl turn for f k = g k g k+1 : Γ k Γ k nd AMII gurntees eh f k is lso tt mp. As wrm-up, Exmple 5.1 demonstrtes the proedure s pplition to n idelly deomposed representtive. We then explin the proedure s steps nd how we used them in Exmple 5.1. This setion onludes with proof of the proedure s vlidity. 1

14 Exmple 5.1. We pply the proedure to show the following idelly deomposed tt mp hs no pnp s. For simpliity, the ltt strutures G i = G(f i ) re shown without lk edges [e j ] onneting the vertex pirs {d j, d j }. Underneth eh ltt struture G i we inluded the illegl turn T i for the genertor g i. We often use nottion y writing e for D 0 (e) where e {, ā,,,, }. G G1 G 0 G G4 g g 1 g g 4 g 5 T0 T 1 T T T4 G5 G6 G7 G8 G9 G10 g g 6 7 g g 8 9 g 10 T T6 T T8 T9 T G 11 G 1 G 1 G 14 G 15 g 11 g g g g T T T T T Proedure Applied: Sine {, } is the only illegl turn for g, ny ipnp would ontin it s its unique illegl turn. We re thus trying to find n ipnp ρ 1 ρ where ρ 1 = e e... nd ρ = e e... re legl pths nd ll e i, e i E(Γ), exept tht the finl edge in either ρ 1 or ρ my e prtil. Sine g 1 () = is the initil supth of g 1 () =, proper neltion would fore ρ to ontin nother edge e fter (Proposition 5. I). Sine lels the red vertex in G 1 (i.e. D 0 () is the unhieved diretion d u 1 ), the edge e would e suh tht D 0(e ) is preimge under Dg 1 of D 0 () (proper neltion requires Dg,1 (e ) = Dg () = D 0 () ut, sine Dg 1 (e ) nnot e the unhieved diretion, nd the only other preimge under Dg of the twie-hieved diretion is the other diretion in the illegl turn T 1 = {, }, nmely the twie-hieved diretion, we must hve Dg 1 (e ) = ). The only preimge under Dg 1 of is. So e =. Thus, so fr, we hve ρ 1 =... nd ρ =... Sine g,1 () = g () is the initil supth of g,1 () = g (), we know ρ 1 would ontin nother edge e fter (Proposition 5. III). Sine lels the red unhieved diretion vertex in G, we know D 0 (e ) nnot e preimge under Dg,1 of. So our est hope is Dg,1 (e ) = Dg (). This n only hppen if Dg,1 (e ) is preimge under Dg of the other diretion in the illegl turn T = {, }, nmely the twie-hieved diretion. There re two suh preimges under Dg,1 (the two diretion in the illegl turn T 0 = {, }): Cse 1 will e where e = nd Cse where e =. We nlyze eh of these se. For Cse 1, suppose e =. Sine g,1 () = g, ()g () is the initil supth of g,1 (āā) = g, ()g (), we know ρ would ontin nother edge e fter. Sine lels the red unhieved diretion vertex in G, we would need D 0 (e ) to e preimge of under Dg (sine T = {, }, this follows s ove). The only suh preimge is. So e =, giving ρ 1 = āā... nd ρ =.... Note tht g 4,1 () = g 4, ()g 4, ()g 4 () nd g 4,1 (āā) = g 4, ()g 4, ()g 4 (). So, sine {, } T 4 (nd ), we ould not hve ρ 1 = āā nd ρ = (Proposition 5. II). Cse 1 ould not our. 14

15 For Cse, suppose e =. Sine g,1 () = g, ()g () is the initil supth of g,1 () = g, ()g (), we know ρ would ontin nother edge e fter. Sine lels the red unhieved diretion vertex in G, we n follow the logi ove nd see tht e would e, giving ρ 1 =... nd ρ =.... Sine g 4,1 () = g 4, ()g 4, ()g 4 () nd g 4,1 () = g 4, ()g 4, ()g 4 (), neltion leves {, }. As ove, we ould not hve ρ 1 =... nd ρ =.... This rules out ll possiilities for ρ 1 ρ nd so g hs no ipnp s, thus no pnp s, s desired. Explntion of Proedure Applied: Let (g 0,..., g n ; G 0,..., G n ) e n ID dmissile omposition with the stndrd nottion. We give the generl proedure for finding ny ipnp ρ = ρ 1 ρ for g = g 0 g n, where ρ 1 = e 1... e m ; ρ = e 1... e m ; e 1,..., e m, e 1,..., e m E(Γ); nd {D 0 (e 1 ), D 0 (e 1 )} = {d 1, d 1 } is the unique illegl turn of ρ. We let ρ 1,k = e 1... e k nd ρ,l = e 1... e l throughout the proedure. After eh step is explined in itlis, we explin its use in Exmple 5.1. (I) Apply g 1, g, et, to e 1 nd e 1 until Dg j,1(e 1 ) = Dg j,1(e 1 ). Either g j,1 (e 1 ) is the initil supth of g j,1 (e 1 ) or vie vers. Without generlity loss (or djust nottion) ssume g j,1(e 1 ) is the supth of g j,1 (e 1 ), so g j,1 (e 1 ) = g j,1 (e 1 )t..., for some edge t. Then ρ must ontin nother edge e. g 1 () = ws the initil supth of g 1 () =. This implied ρ hd n edge fter. (II) Continue omposing genertors g i until either: () one otins g p with g p (ρ,k ) = τ e 1... nd gp (ρ 1,s ) = τ e 1... for legl pth τ (proeed to V), II did not our in Exmple 5.1. When it ours, it mkes n ipnp promising. V my identify n ipnp, if it exists, s does IV. However, IV involves trimming, sine it involves the se where the pth s initil nd finl edges re only prtil edges. IV nd IVd diret one to possily find n ipnp y ontinuing to dd edges. () g j,l (ρ,k ) is supth of g j,l (ρ 1,s ) or vie vers (proeed to III), II ours in oth Cse 1 nd Cse of Exmple 5.1. In Cse 1, III is used to otin ρ 1, = āā nd ρ, =. In Cse, III yields ρ 1, = ā nd ρ, =. () or some g l,1 g p (ρ,k ) = τ γ,k nd g l,1 g p (ρ 1,s ) = τ γ 1,s where T l {D 0 (γ,k ), D 0 (γ 1,s )}. In this se there nnot e n ipnp with ρ,k ρ 1,s s supth. Proeed to VI. In oth Cse 1 nd Cse, fter pplying III to otin ρ 1, nd ρ,, II ours with {D 0 (), D 0 ()} = T 4. In oth ses, τ = g 4, ()g 4, ()g 4 (). In Cse 1, {D 0 (γ,k ), D 0 (γ 1,s )} = {D 0 (), D 0 ()} = {D 0 (), D 0 ()}. In Cse, {D 0 (γ,k ), D 0 (γ 1,s )} = {D 0 (), D 0 ()} = {D 0 (), D 0 ()}. (III) Suppose g j,1 (ρ 1,k ) = g j,1 (ρ,s )t s+1... (or swith e i for e i, ρ 1 for ρ, et). ρ must hve nother edge e s+1 fter ρ,s. There re two ses to onsider: Sine ρ 1,1 =, ρ,1 =, nd g,1 () = g () is the initil supth of g,1 () = g (), we ssumed g,1 (ρ 1,1 ) = g,1 (ρ,1 )t.... In prtiulr, t s+1 =. () If D 0 (t s+1 ) = d u j, then the different possiilities for e s+1 re determined y the diretions d s+1 suh tht T j = {Dg j,1 (d s+1 ), D 0(t s+1 )} where D 0 (e s+1 ) = d s+1. D 0 () leled the red vertex in G 1, implying D 0 () = d u 1, i.e. D 0 (t s+1 ) = d u j. T 1 = {, } implied Dg 1 (d ) =. Sine the only preimge of under Dg 1 ws, we needed d = D 0 (), e =, nd ρ, =. We hit this se gin when determining possiilities for e. () If D 0 (t s+1 ) d u j, the possiilities for e s+1 re ll edges e s+1 suh tht Dg j,1(d 0 (e s+1 )) = D 0(t s+1 ). III did not our in Exmple 5.1. If D 0 () d u 1, we would hve looked for edges e with Dg 1 (d ) =. (Otherwise we ould not hve hd Dg,1 (d ) =.) Sine ρ must e legl, ignore hoies for d s+1 where T 0 = {d s, d s+1 } is the illegl turn for g. Eh remining d s+1 in () or () gives nother prospetive ipnp one must pply the steps to. 15

16 Only e = stisfied Dg 1 (e ) =. However, oth nd (referred to s Cse 1 nd Cse ) gve prospetive diretions nlyzed in finding e. For Cse 1 nd Cse we seprtely ontinued through the steps. (IV) For eh p 1 with g p (ρ,m ) = τ e 1... nd g p (ρ 1,n ) = τ e 1... for some legl pth τ (nd pproprite m nd n), hek if g p # (ρ 1,nρ,m ) ρ 1,n ρ,m or vie vers. () If, for some p 1, g p # (ρ 1,nρ,m ) = ρ 1,n ρ,m, then ρ 1,n ρ,m is the only possile ipnp for g. () For eh p 1 suh tht g p # (ρ 1,nρ,m ) ρ 1,n ρ,m (ontinment proper), proeed to V. () If ρ 1,n ρ,m g p # (ρ 1,nρ,m ) (ontinment proper): The finl ourrene of e n in the opy of ρ 1,n in g p (ρ 1,n ρ,m ) must e from g p (e n ) nd the finl ourrene of e m in the opy of ρ,m in g p (ρ 1,n ρ,m ) must e from g p (e m). Thus, e n nd e m hve fixed points. Reple ρ 1,n ρ,m with ρ 1,n ρ,m, where ρ 1,n ρ,m is ρ 1,nρ,m, ut with e n nd e m repled y prtil edges ending t the fixed points. Repet until some ρ 1,n ρ,m is n ipnp. (d) If we do not hve g p # (ρ 1,nρ,m ) ρ 1,n ρ,m or vie vers for ny 1 p, there is only one possiility for ρ,m ρ 1,n to e supth of n ipnp. It is when g p # (ρ 1,nρ,m ) = γ 1,n γ,m nd either γ 1,n ρ 1,n nd ρ,m γ,m or γ,m ρ,m nd ρ 1,n γ 1,n. In this se, pply V to the side too short. Otherwise, there nnot e n ipnp with ρ,m ρ 1,n s supth (proeed to VI). (VI) Assume g p ( ρ 1,n ρ,m ) ρ 1,n ρ,m (ontinment proper) nd, without generlity loss, g p (ρ 1,n ρ,m )t m+1 ρ 1,n ρ,m for some t m+1. For eh e i suh tht Dg p (D 0 (e i )) = D 0 (t m+1 ) nd {D 0 (e i 1 ), D 0 (e i )} = {D 0 (e 1 ), D 0 (e 1 )} (the illegl turn for g), return to V with ρ,m+1, where e m+1 = e i. (VI) Rule out the other possile supths tht rose vi this proedure (y different hoies of d i, s in III or V). If there re no other possile supths, there re no ipnp s (thus no pnp s) for g. In disovering options for e (fter nlyzing Cse 1), VI sent us k to Cse. Proposition 5.. Let (g 0,..., g n ; G 0,..., G n ) e n ID dmissile omposition with nottion s ove. Then the proedure desried in steps (I)-(VI) determines ll ipnp s for g = g 0 g n. We use the following lemm in the proposition proof (see [Pf1] for proofs). Lemm 5.. [Pf1]. Supths of legl pths re legl.. For trin trks, imges of legl pths nd turns re legl. We lso remind the reder: 1. Sine d u k will e one vertex of the illegl turn T k of G k, T k nnot lso e purple edge in G k.. d u k must e vertex of the red edge [tr k ] of G k.. For eh k, T k is not the red edge in G k, so is not represented y ny edge in G k. Proof. (of Proposition) We egin with n rgument used repetedly. Sine ρ = ρ 1 ρ is n ipnp, ρ 1 nd ρ re oth legl. Sine supths of legl pths re legl nd sine the g k,1 imges of legl pths re legl, g k,1 (e 1... e l ) nd g k,1 (e 1... e l ) re legl pths for eh 1 k n, 1 l m, nd 1 l n. Sine {D 0 (e 1 ), D 0 (e 1 )} is illegl, Dg j,1(e 1 ) = Dg j,1(e 1 ), for some j. We show either g j,1 (e 1 ) is the initil supth of g l,1 (e 1 ) or vie vers. Let d 1 = D 0 (e 1 ) nd d 1 = D 0(e 1 ). Sine {D 0(e 1 ), D 0 (e 1 )} is illegl for g nd T 0 = {d pu 0, dp 0 } is g s only illegl turn, {d 1, d 1 } = {dpu 0, dp 0 }. Without generlity loss suppose d 1 = d pu 0 (nd d 1 = dp 0 ). Sine g 1(e pu 0 ) = e 1 eu 1, we hve g 1(e 1 ) = g 1 (e pu 0 ) = e 1 eu 1 nd g 1(e 1 ) = g 1(e p 0 ) = e 1. So g 1 (e 1 ) is the initil supth of g 1(e 1 ). Sine g j, is n utomorphism, g j, (g 1 (e 1 )) = g j,(e 1 ) is the initil supth of g j, (g 1 (e 1 )) = g j, (e 1 eu 1 ) = g j,(e 1 )g j,(e u 1 ), s desired. Left to show for I is tht ρ must ontin 16