A Remark on Generalized Free Subgroups. of Generalized HNN Groups

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1 Ieraoal Mahemacal Forum o A Remar o Geeralzed Free Subroup o Geeralzed HNN Group R M S Mahmood Al Ho Uvery Abu Dhab POBo 526 UAE raheedmm@yahoocom Abrac A roup ermed eeralzed ree roup a ree produc o cyclc roup A ew cla o roup called eeralzed HNN roup are eeo o HNN roup or a able leer ad a eer 2he eleme he bae I h paper we how ha a ubroup o eeralzed HNN roup eeraed by he couae o he bae ormal ad o quoe a eeralzed ree roup Mahemac Subec Clacao: 20E06 20E07 20E08 Keyword: eeralzed ree roup eeralzed HNN roup Iroduco I h paper he cla o ree roup ha a ree produc o e cyclc roup eeralzed o a ew cla o roup called eeralzed ree roup ha a ree produc o cyclc roup o ay order Furhermorehe cla o HNN roup (Hma Neuma ad Neuma roup) where he power o ay able leer o he bae eeded o a ew cla o roup called eeralzed HNN roup where he power o ome able leer are he bae Th paper dvded o 4 eco I eco 2 we roduce eample o he eeralzed ree roup ad ormulae he ubroup heorem I eco 3 we roduce he cocep o eeralzed HNN roup ad ormulae ome ubroup duced by he bae I eco 4 we how ha a ubroup o a eeralzed HNN roup eeraed by he couae o he bae ormal ad o quoe a eeralzed ree roup

2 504 R M S Mahmood 2 Geeralzed Free Group I well ow ha ree roup are ree produc o e cyclc roup By a he ree produc o cyclc roup o ay order we oba a ew cla o roup ad we call he cla o eeralzed ree roup The ollow are eample o eeralzed ree roup [] The e o eer Z a eeralzed ree roup; [2] Ay e cyclc roup Z o order a eeralzed ree roup; [3] Ay ree roup a eeralzed ree roup; 2 2 [4] The e dhedral roup y = y = C2 C2 a eeralzed ree roup; [5] PSL2 ( Z) 2 C3 a eeralzed ree roup where C2 a cyclc roup o order 2 C 2 = {{ ± M}{ ± N}} adc 3 a cyclc roup o order C 3 = {{ ± M}{ ± N}{ ± L}} where M = N = ad L = 0 0 I well ow ha a ubroup o a ree roup ree ([3 Prop 2 p 8]) Th reul ca be ealy eeralzed o eeralzed ree roup a ollow The ubroup heorem or ree produc o roup ( [ 3 Prop 36 p 20]) ae ha G he ree produc o he roup G where ru over a de e I ad H a ubroup o G he H he ree produc o a ree roup oeher wh roup ha are couae o ubroup o he ree acor G o G Sce ubroup o cyclc roup are cyclch lead he ma reul o h eco Theorem 2 Subroup o eeralzed ree roup are eeralzed ree roup 3 Geeralzed HNN Group The Hma Neuma ad Neuma roup deoed HNN roup appeared everal boo See [3 Chaper IV] HNN roup ay he codo ha he power o ay able leer o ay couae o he bae I [2] Khaar ad Mahmood eeralzed he cla o HNN roup o a ew cla o roup called eeralzed HNN roup where he power o ome able leer a couae o he bae a ollow Le G be a roup ad I ad J be wo de e uch ha I J = ad I J Le { A : I} { B : I} ad { C : J} be amle o ubroup o G For each I le φ : A B be a oo omorphm ad

3 Remar o eeralzed ree ubroup 505 or each J le α : C C be a auomorphm uch ha c α c ed by 2 a er auomorphm deermed by C ad α Tha α ( c ) ad α (c) c c or all c C The roup G* deermed by he ollow preeao G* = e ( G) l rel (G) A = B C I J called a eeralzed HNN roup o bae G ad aocaed par ( A B ) ad ( C C ) o ubroup o G The ymbol I J called a able leer The G Furhermore p q I J p q he p q The oao o he preeao o G* ca be eplaed a ollow () e( G) rel( G) ad or ay preeao o G where e(g) a e o eera ymbol ad rel(g) a e o relao o he preeao o G; (2) A = B ad or he e o relao w(a) = w( φ (a)) where w(a) ad w( φ ( a)) are word e(g) o value a ad φ ( a) repecvely where a ru over a e o eeraor o A ; (3) C ad or he e o relao w( c) = w( α ( c)) where w(c) ad w( α ( c)) are word e(g) o value c ad α (c) repecvely where c ru over a e o eeraor oc ; (4) ad or he o relao = w(c ) where w(c ) a word e(g) o value c The embedd heorem ad Bro` lemma or eeralzed HNN roup obaed [2] ca be aed a ollow Lemma 3 G embedded G* ad every eleme o G* ca be e e2 e wre a a reduced word o G* Tha = o 2 2 where G e = ± I J or = uch ha coa o ubword o he ollow orm [] a a A or [2] b b B or e δ [3] c c C e δ = ± or [4] or ome J Noao I M he ubroup o G coa he ubroup ad C J le M * be he roup o he preeao A B I

4 506 R M S Mahmood e ( M ) l rel (M) A = B C I J I clear ha M * a eeralzed HNN roup o bae M ad aocaed par ( A B ) ad ( C C ) o ubroup o M The ma reul o h eco are he ollow Theorem 32 Le M be he ubroup o G coa he ubroup A B ~ I ad C J ad M = 2 M be he ubroup o G* eeraed by ~ ~ 2 ad M The M = M * M G = M ad G* he ree produc o he roup G ad M* wh a amalamao ubroup M Tha ; G * = G M * Proo Le e( M ) = { : M } be he e o eera ymbol ad rel( M )= { = : M } be he e o relao o he preeao o M Le F be he ree roup o bae he eera ymbol o M * The he bae S o F co o he eera ymbol o M ad he able leer I J Le ψ: S M be he uco ve by ψ ( ) = ad ψ( ) = I J The here e a uque homomorphm Ψ : F M ~ ay he codo ha Ψ S = ψ So Ψ ( ) = ad Ψ ( ) = I J The clear ha Ψ ae he relao o M * The here e a uque homomorphm Ψ ~ : M * M ~ ay Ψ ~ () = Ψ ( ) = ad Ψ ~ ( ) = I J Sce M ~ eeraed by he eeraor o M ad by I J hereore Ψ ~ a epmomorphm Now we how ha Ψ ~ ecve Le be a eleme o M * We eed o how ha Ψ ~ () e e2 e By Lemma 3 ca be wre a a reduced word = o M * where G e = ± I J or = The Ψ ~ e e2 e () = o 2 2 ~ So M = M * M * wh a amalamao ubroup M Sce * o 2 Th mple ha Ψ ~ a omorphm Now we how ha G* he ree produc o he roup G ad M a eeralzed HNN rouphereore by Lemma 3 M embedded a a ubroup M M * The here a omorphm rom M o M deoed m m m M The by [2 Theo 43 p 99] G M * ha he preeao e ( G) e( M ) l rel(g)rel (M) A where I J Hece he preeao M = B C e ( G) l rel (G) A = B C I J Th mple ha G * = G M * Th complee he proo M M 2

5 Remar o eeralzed ree ubroup Ma Reul The ollow lemma [ Theo 2 p 7] eeded or he proo o he ma reul o h paper Lemma 4 Le he roup G have he preeao G = S R ad H be a ormal ubroup o G eeraed by he e P The he quoe roup G H ha he preeaog H = S R P where P he e o relao p = p P ad p a word he e S o eera ymbol o G o value p I he ollow G* he eeralzed HNN roup e ( G) l rel (G) A = B C I J o bae G ad aocaed par ( A B ) ad ( C C ) o ubroup o G Propoo 42 The roup K = roup l = I J a eeralzed ree Proo I clear ha he roup K a ree produc o he e cyclc roup C Z eeraed by I ad o e cyclc roup C o order eeraed by J where Z he e o all eer Th mple ha K a eeralzed ree roup The ma reul o h paper he ollow heorem Theorem 43 Le G* be he eeralzed HNN roup e ( G) l rel (G) A = B C I J o bae G ad aocaed par ( A B ) ad ( C C ) o ubroup o G Le H be he ubroup o G* eeraed by he couae o G The [] H a ormal ubroup o G* ; [2] G * H = I J [3] G * H l ; a eeralzed ree roup Proo [] For G* le G =G The H = G ; G * For h H ad G* we have h = hh2 h where h G G * or = 2 ad h = h h2 h The h G = G H = 2 Th mple ha h H Coequely H a ormal ubroup o G* [2] Le e(g) be he e o eera ymbol

6 508 R M S Mahmood e(g) = { : G} ad rel(g) be he e o relao rel(g) = { = ; G} o he preeao o G For G* le w be a word he eera ymbol o G* o value Sce H eeraed by he e o eleme o he orm hereore by Lemma 4 G * H ha he preeao = w w = A = B C c = I J G Sce he relao w w = G are coequece o he relao = hereore by he Teze raormao T 2 [ p 49] G * H ha he preeao G * H = = = A = B C = c I J G By he Teze raormao T 4 [ p 50] we delee he ymbol rom he e o eera ymbol o G* ad delee he relao = rom he e o relao o G* ad ubue by he relao A = B C ad The A = B ad C become rval relao ad ca be deleed ad become he relao = Coequely G * H ha he preeao l = I J [3] Follow rom Propoo 42 Th complee he proo By a J = o he preeao o G* yeld ha G* a HNN roup Th mple he ollow corollary Corollary 44 Le G* = e ( G) l rel (G) A = B I be he HNN roup o bae G ad aocaed par ( A B ) o ubroup o G Le H be he ubroup o G* eeraed by he couae o G The [] H a ormal ubroup o G* ; [2] G * H I ; [3] G * H a ree roup I [3 Theorem 5 pae ] how ha G = X r a oe-relaor roup where r cyclcally reduced ad coa a lea wo dere leer rom Xhe G ca be embedded a HNN roup e ( K) rel( K) U = V where he bae K a oe-relaor roup K X r where r cyclcally reduced ad r horer ha r ad he aocaed ubroup U ad V o he bae K are omorphc ree roup Th lead he ollow corollary o Corollary 44

7 Remar o eeralzed ree ubroup 509 Corollary 45 Le G ad K be he roup deed above Le H be he ubroup o G eeraed by he couae o K The H a ormal ubroup o G ad G H a ree roup Acowledeme The auhor would le o ha he reeree or h cere evaluao ad corucve comme whch mproved he paper coderably Reerece [] W Mau A Karra ad D Solar Combaoral roup heory Dover PublIc New Yor 976 [2] M I Khaar ad R M S Mahmood O qua HNN roup Kuwa J Sc E 29(2) 2002 [3] R C Lydo P E Schupp Combaoral Group Theory prer Verla Berl New Yor 977 Receved: Ocober 2009