Ôèáîíà è Îáîáùåíèÿ Êîñû m ñëîâ Ãðóïïû ñ öèêëè åñêèì êîïðåäñòàâëåíèåì è òðåõìåðíûå ìíîãîîáðàçèÿ Âåñíèí À.Þ. Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë.Ñîáîëåâà ÑÎ ÐÀÍ

Ãðóïïû ñ öèêëè åñêèì êîïðåäñòàâëåíèåì è òðåõìåðíûå ìíîãîîáðàçèÿ Âåñíèí À.Þ. Èíñòèòóò ìàòåìàòèêè èì. Ñ.Ë.Ñîáîëåâà ÑÎ ÐÀÍ Ìàëüöåâñêèå òåíèÿ Íîâîñèáèðñê, 11 íîÿáðÿ 2008 ã.

1 Ãåîìåòðèÿ è òîïîëîãèÿ ãðóïï Ôèáîíà è Ãðóïïû Ôèáîíà è Ìíîãîîáðàçèÿ Ôèáîíà è Öèêëè åñêèé àâòîìîðôèçì è öèêëè åñêîå íàêðûòèå 2 Îáîáùåííèÿ ãðóïï Ôèáîíà è Ãðóïïû Ñèðàäñêè è Êàâèêêèîëè Õåãåíáàðòà Ðåïîâøà Îáîáùåííûå ãðóïïû Ôèáîíà è Äðîáíûå ãðóïïû Ôèáîíà è Ãðóïïû ñ öèêëè åñêèì ïðåäñòàâëåíèåì 3 Ñâÿçü ñ ãðóïïàìè êîñ 4 m-ñëîâåñòíîå öèêëè åñêîå êîïðåäñòàâëåíèå

Ãðóïïû Ôèáîíà è: F (2, m) = x 1,..., x m x i x i+1 = x i+2, i = 1,..., m.

Ãðóïïû Ôèáîíà è: F (2, m) = x 1,..., x m x i x i+1 = x i+2, i = 1,..., m. Conway 1965: ÿâëÿåòñÿ ëè ãðóïïà F (2, 5) öèêëè åñêîé ïîðÿäêà 11? ïðè êàêèõ m ãðóïïà F (2, m) êîíå íà?

Ãðóïïû Ôèáîíà è: F (2, m) = x 1,..., x m x i x i+1 = x i+2, i = 1,..., m. Conway 1965: ÿâëÿåòñÿ ëè ãðóïïà F (2, 5) öèêëè åñêîé ïîðÿäêà 11? ïðè êàêèõ m ãðóïïà F (2, m) êîíå íà? Conway 1967, Brunner 1974, Havas 1976, Chalk Johnson 1976, Newman 1988, Thomas 1989: F (2, m) êîíå íà m = 1, 2, 3, 4, 5, 7. À èìåííî, F (2, 3) = Q 8, F (2, 4) = Z 5, F (2, 5) = Z 11, F (2, 7) = Z 29.

Åñëè m íå åòíî, òî ãðóïïà Ôèáîíà è ñîäåðæèò êðó åíèå. F (2, m) = x 1,..., x m x i x i+1 = x i+2, i = 1,..., m Ðàññìîòðèì ýëåìåíò u = x 1 x 2... x m, òîãäà u 2 = (x 1 x 2 )(x 3 x 4 ) (x m x 1 )(x 2 x 3 ) (x m 1 x m ) = x 3 x 5 x m x 2 x 4 x 1 = (x2 1 x 4)(x4 1 x 6) (xm 1 1 x 1)(x1 1 x 3)(x3 1 x 5) (xm 1 x 2 ) = 1. Âûïîëíåíèå u = 1 ïðèâåëî áû ê òîìó, òî âñå ïîðîæäàþùèå x 1,..., x m ïîïàðíî êîììóòèðóþò.

Â ñàìîì äåëå, åñëè u = x 1 x 2 x 3 x 4 x 5 x 6 x m = 1, òî x3 2 x 4 x 5 x 6 x m = 1 x 3 x5 2 x 6 x m = 1 x 3 x 5 x7 2 x m = 1 x 3 x 5 x 7 xm 2 = 1 (x2 1 4)(x4 1 6)(x6 1 8) (xm 1 1 1)x m = 1 x2 1 x 1 x m = 1 Ñðàâíèâàÿ ñ x m x 1 = x 2, ïîëó àåì, òî x 1 è x m êîììóòèðóþò. Àíàëîãè íî, êîììóòèðóþò x i è x i+1, è äàëåå, x i è x j. Ïîñêîëüêó ïðè íå åòíîì m 9 ãðóïïà F (2, m) áåñêîíå íà, à åå àáåëèçàòîð êîíå åí, òî u 1.

Helling Kim Mennicke 19881998: ãðóïïà Ôèáîíà è F (2, 2n), n 2, èçîìîðôíà ôóíäàìåíòàëüíîé ãðóïïå çàìêíóòîãî îðèåíòèðóåìîãî 3-ìíîãîîáðàçèÿ. Äîê-âî: ÿâíàÿ êîíñòðóêöèÿ ôóíäàìåíòàëüíîãî ìíîãîãðàííèêà P n äëÿ ãðóïïû F (2, 2n). Ãðàíèöà ìíîãîãðàííèêà P n ñîñòîèò èç 4n òðåóãîëüíûõ ãðàíåé è P n îáëàäàåò öèêëè åñêîé ñèììåòðèåé ïîðÿäêà n. P 5 èêîñàýäð.

Ôóíäàìåíòàëüíûé ìíîãîãðàííèê P 4 : åñëè i íå åòíî, òî x i : QP i+1 P i+3 P i+2 P i+3 P i+4, åñëè i åòíî, òî x i : RP i+1 P i+3 P i+2 P i+3 P i+4. Q F 7 F 1 F 3 F 5 P 8 P 2 P 4 P 6 P 8 F F 6 F 5 F 8 F 7 F 2 F 1 F 4 3 P 7 P 1 P 3 F P 5 P 7 6 F 8 F 2 F 4 R

Ãåîìåòðèÿ: P n X n ãäå X n = S 3, n = 2, E 3, n = 3, H 3, n 4. F (2, 2n) ðåàëèçóåòñÿ êàê ãðóïïà èçîìåòðèé ïðîñòðàíñòâà X n, ïîðîæäåííàÿ ïîïàðíûìè îòîæäåñòâëåíèÿìè ãðàíåé P n. Òðåõìåðíîå ìíîãîîáðàçèå M n = X n /F (2, 2n), n 2, íàçûâàåòñÿ ìíîãîîáðàçèåì Ôèáîíà è. M 2 = S 3 /F (2, 4) = S 3 /Z 5 = L(5, 2) ëèíçîâîå ïðîñòðàíñòâî; M 3 = E 3 /F (2, 6) åâêëèäîâî ìíîãîîáðàçèå; M n = H 3 /F (2, 2n), n 4 ãèïåðáîëè åñêîå ìíîãîîáðàçèå.

Kuiper 1989: âåðíî ëè, òî ìíîãîîáðàçèå Ôèáîíà è M 4 = H 3 /F (2, 8) èìååò íàèìåíüøèé îáú¼ì ñðåäè âñåõ çàìêíóòûõ îðèåíòèðóåìûõ ãèïåðáîëè åñêèõ 3-ìíîãîîáðàçèé? Â. 1991: Íåò. volm 4 = 2.02... > 0.94... = volm FMW.

Kuiper 1989: âåðíî ëè, òî ìíîãîîáðàçèå Ôèáîíà è M 4 = H 3 /F (2, 8) èìååò íàèìåíüøèé îáú¼ì ñðåäè âñåõ çàìêíóòûõ îðèåíòèðóåìûõ ãèïåðáîëè åñêèõ 3-ìíîãîîáðàçèé? Â. 1991: Íåò. volm 4 = 2.02... > 0.94... = volm FMW. Thurston 1978: ñóùåñòâóåò ëè êîìïàêòíîå ãèïåðáîëè åñêîå 3-ìíîãîîáðàçèå, îáú¼ì êîòîðîãî ðàâåí îáú¼ìó íåêîìïàêòíîãî? Â. Ìåäíûõ 1994: Äà. Ïîëó åíû òî íûå ôîðìóëû äëÿ îáú¼ìîâ ãèïåðáîëè åñêèõ ìíîãîîáðàçèé Ôèáîíà è. Èç íèõ ñëåäóåò, òî äëÿ n 2 îáú¼ì êîìïàêòíîãî ìíîãîîáðàçèÿ M 2n ðàâåí îáú¼ìó íåêîìïàêòíîãî.

P n îáëàäàåò öèêëè åñêîé ñèììåòðèåé ρ ïîðÿäêà n âðàùåíèåì âîêðóã îñè QR: ρ : F i F i+2, ρ : F i F i+2 ρ èíäóöèðóåò àâòîìîðôèçì ãðóïïû F (2, 2n) òàêîé, òî ρ : x i x i+2 = ρ 1 x i ρ. Äëÿ ãðóïïû Γ n = F (2, 2n), ρ ôóíäàìåíòàëüíûì ìíîãîãðàííèêîì ÿâëÿåòñÿ 1 n äîëüêà ìíîãîãðàííèêà P n.

Ðåáðà äîëüêè ðàçáèâàþòñÿ íà êëàññû ýêâèâàëåíòíûõ îòíîñèòåëüíî äåéñòâèÿ ãðóïïû Γ n : Q P i+1 P i+3 R P i+2 P i+4

Ïåðâûé öèêë ýêâèâàëåíòíûõ ðåáåð: x QP i ρx 1 i+1 i+1 Pi+2 P i+3 îòêóäà ρ 1 x i ρ = x i x i+1. x 1 i ρ P i+2 P i+4 QP 1 i+3 QP i+1,

Ïåðâûé öèêë ýêâèâàëåíòíûõ ðåáåð: x QP i ρx 1 i+1 i+1 Pi+2 P i+3 îòêóäà ρ 1 x i ρ = x i x i+1. Âòîðîé öèêë ýêâèâàëåíòíûõ ðåáåð: x i+1 ρ RP 1 i+4 x 1 i ρ P i+2 P i+4 QP 1 i+3 QP i+1, ρx i+2 ρ P i+1 P 2 ρ x 1 i+1 i+3 P i+1 P i+2 P i+3 P i+4 îòêóäà ρ 1 x i+1 ρ = x i+1 x i x i+1. RP i+2 ρ RP i+4,

Ïåðâûé öèêë ýêâèâàëåíòíûõ ðåáåð: x QP i ρx 1 i+1 i+1 Pi+2 P i+3 îòêóäà ρ 1 x i ρ = x i x i+1. Âòîðîé öèêë ýêâèâàëåíòíûõ ðåáåð: x i+1 ρ RP 1 i+4 x 1 i ρ P i+2 P i+4 QP 1 i+3 QP i+1, ρx i+2 ρ P i+1 P 2 ρ x 1 i+1 i+3 P i+1 P i+2 P i+3 P i+4 îòêóäà ρ 1 x i+1 ρ = x i+1 x i x i+1. Òðåòèé öèêë: QR ρ QR, îòêóäà ρ n = 1. RP i+2 ρ RP i+4,

Ñîãëàñíî òåîðåìå Ïàóíêàðå î ôóíäàìåíòàëüíîì ìíîãîãðàííèêå, ãðóïïà Γ n èìååò êîïðåäñòàâëåíèå: ρ, x i, x i+1 ρ n = 1, ρ 1 x i ρ = x i x i+1, ρ 1 x i+1 ρ = x i+1 x i x i+1. Èç ρ 1 x i+1 ρ = x i+1 ρ 1 x i ρ âûðàçèì x i : Òàêèì îáðàçîì, x i = ρx 1 i+1 ρ 1 x i+1. x 1 i+1 ρ 1 x i+1 ρ = ρx 1 i+1 ρ 1 x i+1 x i+1.

Ïóñòü b = x i+1 ρ 1, òîãäà Γ n = ρ, b ρ n = b n = 1, ρ 1 [b, ρ] = [b, ρ] b, ãäå [b, ρ] = b 1 ρ 1 bρ.

Ïóñòü b = x i+1 ρ 1, òîãäà Γ n = ρ, b ρ n = b n = 1, ρ 1 [b, ρ] = [b, ρ] b, ãäå [b, ρ] = b 1 ρ 1 bρ. Õîðîøî èçâåñòíî, òî α, β β 1 [α, β] = [α, β] α = π 1 (S 3 \ K) ãðóïïà óçëà âîñüìåðêà K:

Hilden Lozano Montesinos 1992: Ìíîãîîáðàçèå Ôèáîíà è M n, n 2, ÿâëÿåòñÿ n-ëèñòíûì öèêëè åñêèì íàêðûòèåì S 3, ðàçâåòâëåííûì íàä óçëîì âîñüìåðêà.

Maclachlan 1995: Åñëè m íå åòíî, òî ãðóïïà F (2, m) íå ìîæåò áûòü ôóíäàìåíòàëüíîé ãðóïïîé ãèïåðáîëè åñêîãî 3-îðáèôîëäà êîíå íîãî îáúåìà.

Ãðóïïû Ñèðàäñêè S(n) = x 1,..., x n x i x i+2 = x i+1, i = 1,..., n

Ãðóïïû Ñèðàäñêè S(n) = x 1,..., x n x i x i+2 = x i+1, i = 1,..., n Cavicchioli Hegenbarth Kim 1998: ãðóïïà Ñèðàäñêè S(n) ÿâëÿåòñÿ ôóíäàìåíòàëüíîé ãðóïïîé 3-ìíîãîîáðàçèÿ, êîòîðîå åñòü n-ëèñòíîå öèêëè åñêîå íàêðûòèå S 3, ðàçâåòâëåííîå íàä óçëîì òðèëèñòíèê.

Cavicchioli Hegenbarth Repov s 1998: G n (m, k) = x 1, x 2,..., x n x i x i+m = x i+k, i = 1,..., n. ßâëÿåòñÿ ëè ôóíäàìåíòàëüíîé ãðóïïîé 3-ìíîãîîáðàçèÿ?

Cavicchioli Hegenbarth Repov s 1998: G n (m, k) = x 1, x 2,..., x n x i x i+m = x i+k, i = 1,..., n. ßâëÿåòñÿ ëè ôóíäàìåíòàëüíîé ãðóïïîé 3-ìíîãîîáðàçèÿ? Áàðäàêîâ - Â. 2003: åñëè n íå åòíî, k m åòíî è (m 2k, n) = 1, òî ãðóïïà G n (m, k) íå ìîæåò áûòü ãðóïïîé ãèïåðáîëè åñêîãî 3-îðáèôîëäà (â àñòíîñòè, 3-ìíîãîîáðàçèÿ) êîíå íîãî îáúåìà.

Johnson 1974: ãðóïïû F (r, m) = x 1, x 2,..., x m x i x i+r 1 = x i+r, i = 1,..., m, íàçâàíû îáîáùåííûìè ãðóïïàìè Ôèáîíà è.

Johnson 1974: ãðóïïû F (r, m) = x 1, x 2,..., x m x i x i+r 1 = x i+r, i = 1,..., m, íàçâàíû îáîáùåííûìè ãðóïïàìè Ôèáîíà è. Johnson Wamsley Wright 1974, Thomas 1989: ãðóïïû F (r, 2) êîíå íû; åñëè m > 2r + 1, òî ãðóïïà F (r, m) áåñêîíå íà, çà èñêëþ åíèåì ñëó àÿ F (7, 2) è, âîçìîæíî, F (3, 9).

Thomas 1991: ïîðÿäêè ãðóïï F (r, m): r\m 2 3 4 5 6 7 8 9 10 2 1 8 5 11 29 3 8 2 22 1512?? 4 3 63 3???? 5 24 624 4?? 6 5 5 125 7775 5?? 7 48 342? 7 6 1 6? 8 7 7? 8 7 1 7? 9 80 8 6560? 9 8 1 8 10 9 999 4905 9?? 10 9 1 9

Campbell Robertson 1975: F (r, m, k) = x 1,..., x m x i x i+1 x i+r 1 = x i+r+k 1, ãäå i = 1,..., m.

Campbell Robertson 1975: F (r, m, k) = x 1,..., x m x i x i+1 x i+r 1 = x i+r+k 1, ãäå i = 1,..., m. Åñëè d = (m, k 1) è F (r, d) êîíå íà, òî F (r, m, k) êîíå íà.

Campbell Robertson 1975: F (r, m, k) = x 1,..., x m x i x i+1 x i+r 1 = x i+r+k 1, ãäå i = 1,..., m. Åñëè d = (m, k 1) è F (r, d) êîíå íà, òî F (r, m, k) êîíå íà. Szczepa nski V. 2000: åñëè r åòíî, m íå åòíî è (m, r + 2k 1) = 1, òî ãðóïïà F (r, m, k) íå ìîæåò áûòü ôóíäàìåíòàëüíîé ãðóïïîé ãèïåðáîëè åñêîãî 3-îðáèôîëäà êîíå íîãî îáúåìà.

Campbell Robertson 1975: F (r, m, k) = x 1,..., x m x i x i+1 x i+r 1 = x i+r+k 1, ãäå i = 1,..., m. Åñëè d = (m, k 1) è F (r, d) êîíå íà, òî F (r, m, k) êîíå íà. Szczepa nski V. 2000: åñëè r åòíî, m íå åòíî è (m, r + 2k 1) = 1, òî ãðóïïà F (r, m, k) íå ìîæåò áûòü ôóíäàìåíòàëüíîé ãðóïïîé ãèïåðáîëè åñêîãî 3-îðáèôîëäà êîíå íîãî îáúåìà. Äëÿ êàæäîãî m ãðóïïà F (m 1, m, 1) ÿâëÿåòñÿ ôóíäàìåíòàëüíîé ãðóïïîé ïðîñòðàíñòâà Çåéôåðòà Σ m = (0 o 0 1 (2, 1), (2, 1),..., (2, 1) ). }{{} m

H(r, m, s) = x 1,..., x m x i x i+1 x i+r 1 = x i+r x i+r+s 1, ãäå i = 1,..., m.

H(r, m, s) = x 1,..., x m x i x i+1 x i+r 1 = x i+r x i+r+s 1, ãäå i = 1,..., m. Szczepa nski V. 2000: äëÿ k 2 ãðóïïà H(k, 2k 1, k 1) ÿâëÿåòñÿ ôóíäàìåíòàëüíîé ãðóïïîé çàìêíóòîãî îðèåíòèðóåìîãî 3-ìíîãîîáðàçèÿ, êîòîðîå åñòü (2k 1)ëèñòíîå öèêëè åñêîå íàêðûòèå S 3 ðàçâåòâëåííîå íàä òîðè åñêèì (2k 1, 2)óçëîì.

Maclachlan 1995: äëÿ öåëîãî k 1 îïðåäåëèì F k (2, m) = x 1, x 2,..., x m x i x k i+1 = x i+2, i = 1,..., m. Â àñòíîñòè, ïðè k = 1 ïîëó àåì ãðóïïó Ôèáîíà è. Ýòî îáîáùåíèå ãðóïï Ôèáîíà è ñîîòâåòñòâóåò îáîáùåíèþ èñåë Ôèáîíà è: a i+2 = a i + k a i+1 Ìíîãîîáðàçèÿ Ìàêëà ëàíà: ñåðèÿ çàìêíóòûõ îðèåíòèðóåìûõ 3-ìíîãîîáðàçèé M k n òàêèõ, òî π 1 (M k n ) = F k (2, 2n).

Ðàññìîòðèì äðîáíîå îáîáùåíèå èñåë Ôèáîíà è: a i+2 = a i + k l a i+1, (k, l) = 1. Äëÿ âçàèìíî ïðîñòûõ k è l, öåëîãî m > 2 ðàññìîòðèì ãðóïïó F k/l (2, m) = x 1, x 2,..., x m x l i x k i+1 = x l i+2, i = 1,..., m. Ãðóïïó F k/l (2, m) íàçîâåì äðîáíîé ãðóïïîé Ôèáîíà è. Â àñòíîñòè, ïðè k/l = 1 ïîëó àåì ãðóïïó Ôèáîíà è, à ïðè l = 1 ãðóïïó Ìàêëà ëàíà.

Kim V. 1998: ïðè n 2 ãðóïïà F k/l (2, 2n) ÿâëÿåòñÿ ôóíäàìåíòàëüíîé ãðóïïîé çàìêíóòîãî îðèåíòèðóåìîãî 3-ìíîãîîáðàçèÿ. Ä-âî: êîíñòðóêòèâíîå. Îáîçíà èì åðåç Mn k/l äðîáíîå ìíîãîîáðàçèå Ôèáîíà è: π 1 (Mn k/l ) = F k/l (2, 2n).

Òîïîëîãè åñêàÿ åñòåñòâåííîñòü äðîáíîãî îáîáùåíèÿ: Kim V. 1996: Äðîáíîå ìíîãîîáðàçèå Ôèáîíà è Mn 1/l ÿâëÿåòñÿ nëèñòíûì öèêëè åñêèì íàêðûòèåì S 3, ðàçâåòâëåííûì íàä ðàöèîíàëüíûì óçëîì K(2l + 1 2l ) Ðàöèîíàëüíûå óçëû K(2 + 1 2 ), K(4 + 1 4 ) è K(6 + 1 6 ).

Ãðóïïà G îáëàäàåò öèêëè åñêèì êîïðåäñòàâëåíèåì, åñëè îíà ìîæåò áûòü çàäàíà â âèäå G n (w) = x 1, x 2,..., x n w = 1, η(w) = 1,..., η n 1 (w) = 1, ãäå w ñëîâî â àëôàâèòå X = {x 1 ±1, x 2 ±1,..., x n ±1 }, à η àâòîìîðôèçì ñâîáîäíîé ãðóïïû F n = F n (x 1,..., x n ), îïðåäåëåííûé íà ïîðîæäàþùèõ: η(x i ) = x i+1, i = 1,..., n. Åñëè w = x 1 x 2 x3 1, òî G n(w) ãðóïïà Ôèáîíà è. Åñëè w = x 1 x 3 x2 1, òî G n(w) ãðóïïà Ñèðàäñêè.

1. Êîãäà ãðóïïà G n (w) êîíå íà?

1. Êîãäà ãðóïïà G n (w) êîíå íà? 2. Êîãäà ãðóïïà G n (w) ÿâëÿåòñÿ ãðóïïîé 3-ìíîãîîáðàçèÿ?

1. Êîãäà ãðóïïà G n (w) êîíå íà? 2. Êîãäà ãðóïïà G n (w) ÿâëÿåòñÿ ãðóïïîé 3-ìíîãîîáðàçèÿ? 3. Îïèñàòü òîïîëîãè åñêîå äåéñòâèå àâòîìîðôèçìà η.

Artin 1923: êîïðåäñòàâëåíèå ãðóïïû êîñ B 4 íà 4 íèòÿõ ïîðîæäàþùèå: σ 1, σ 2, σ 3 σ 1 σ 2 σ 3 ñîîòíîøåíèÿ: σ 2 σ 1 σ 2 = σ 1 σ 2 σ 1, σ 3 σ 2 σ 3 = σ 2 σ 3 σ 2, σ 3 σ 1 = σ 1 σ 3.

Ãîðèí Ëèí 1969, Áîêóòü Â. 2002 ïîðîæäàþùèå: a, b, t 1, t 2, t ãäå a = σ 1 σ 2 σ1 1 σ 3σ2 1 σ 1 1, b = σ 3σ1 1, t 1 = σ 1 σ 2 σ 2 1, t 2 = σ 2 σ 1 1, t = σ 1. a b t 1 t 2 t

ñîîòíîøåíèÿ: t1 1 1 = b, t1 1 1 = ba 1 b 2, t2 1 2 = b 1 a, t2 1 2 = ba 1 b t 1 at = ba, t 1 bt = b, t 1 t 1 t = t 2, t 1 t 2 t = t 2 t 1 1 B 4 êàê áàøíÿ HNN-ðàñøèðåíèé ñâîáîäíîé ãðóïïû: a, b a, b, t 1, t 2 B 4 = a, b, t 1, t 2, t.

Ðàññìîòðèì äåéñòâèå t 1 íà a, b : t 1 1 at 1 = b, t 1 1 bt 1 = ba 1 b 2. Îáîçíà èì z 0 = b è z i = t1 i z 0t1 i äëÿ i Z. Òîãäà a = z 1 è z 1 = z 0 z 1 1 z2 0. Ñëåäîâàòåëüíî äëÿ i Z èìååì z i+1 z 1 i z 2 i+1 = z i+2, è åñòåñòâåííî âîçíèêàåò áåñêîíå íî-ïîðîæäåííàÿ ãðóïïà H = z i, i Z (z 1 i z i+1 )z i+1 = (z 1 i+1 z i+2), i Z.

Óñå åíèå ýòîé ãðóïïû èìååò âèä: H n = z 1,..., z n (z 1 i z i+1 )z i+1 = (z 1 i+1 z i+2), i = 1,..., n, ãäå âñå èíäåêñû áåðóòñÿ ïî ìîäóëþ n. Îáîçíà èì x 2i 1 = z i è x 2i = z 1 i z i+1 äëÿ i = 1,..., n. Òîãäà ïðèâåäåííûå âûøå ñîîòíîøåíèÿ ïðèìóò âèä x 2i = x 1 2i 1 x 2i+1 è x 2i x 2i+1 = x 2i+2. Ñëåäîâàòåëüíî, ãðóïïà H n èçîìîðôíà ãðóïïå Ôèáîíà è F (2, 2n) = x 1,..., x 2n x i x i+1 = x i+2, i = 1,..., 2n.

Ðàññìîòðèì äåéñòâèå t ñîïðÿæåíèåì íà ãðóïïå t 1, t 2 : t 1 t 1 t = t 2, t 1 t 2 t = t 2 t 1 1. Îáîçíà èì x 0 = t 2 è x i = t i x 0 t i äëÿ i Z. Â àñòíîñòè, x 1 = t 1 è x 1 = t 2 t1 1 = x 0 x1 1. Ñëåäîâàòåëüíî, x i x i+2 = x i+1, è åñòåñòâåííî âîçíèêàåò áåñêîíå íî-ïîðîæäåííàÿ ãðóïïà G = x i, i Z x i x i+2 = x i+1, i Z Óñå åíèå G n ýòîé ãðóïïû åñòü ãðóïïà Ñèðàäñêè S(n) = x 1,..., x n x i x i+2 = x i+1, i = 1,..., n. Çäåñü x i = σ i 1σ 2 σ (i+1) 1.

4. Ñâÿçàíû ëè äðóãèå öèêëè åñêè êîïðåäñòàâèìûå ãðóïïû ñ áàøåííûìè ïðåäñòàâëåíèÿìè ãðóïï êîñ?

Ïóñòü F mn - ñâîáîäíàÿ ãðóïïà ñ ïîðîæäàþùèìè X = {x i,j 1 i m, 1 j n} Çàôèêñèðóåì m ñëîâ w 1,..., w m èç F mn. Ðàññìîòðèì öèêëè åñêèé àâòîìîðôèçì θ n : F mn F mn, θ n (x i,j ) = x i,j+1 äëÿ êàæäîãî i = 1,..., m è j = 1,..., n (âòîðîé èíäåêñ áåðåòñÿ ïî ìîäóëþ n).

Îïðåäåëèì ìíîæåñòâî ñëîâ R = {r i,j r i,j = θ j 1 n (w i ), 1 i m, 1 j n}. Êîïðåäñòàâëåíèå G n (w 1,..., w m ) = X R áóäåì íàçûâàòü m-ñëîâåñòíûì öèêëè åñêèì êîïðåäñòàâëåíèåì.

Cristofori Mulazzani V. 2007: êàæäîå n-ëèñòíîå ñòðîãî-öèêëè åñêîå ðàçâåòâëåííîå íàêðûòèå M íàä (g, 1)-óçëîì K N äîïóñêàåò õåãîðîâî ñïëåòåíèå ðîäà gn ñ öèêëè åñêîé ñèììåòðèåé ïîðÿäêà n, êîòîðîå èíäóöèðóåò g-ñëîâåñòíîå öèêëè åñêîå êîïðåäñòàâëåíèå ôóíäàìåíòàëüíîé ãðóïïû ìíîãîîáðàçèÿ M. Ä-âî ïðèâîäèò ê àëãîðèòìó îïèñàíèÿ êîïðåäñòàâëåíèÿ.

Ïðèìåð. Ïóñòü Ãðóïïà w 1 = x 2,1 x r 1,1 x 1 2,2, w 2 = x 1,1 x p 2,2 x 1 1,2. G n (w 1, w 2 ) = x 1,1, x 1,2,..., x 1,n, x 2,1, x 2,2,..., x 2,n x 2,j x r 1,j = x 2,j+1, x 1,j x p 2,j+1 = x 1,j+1, j = 1,..., n åñòü ôóíäàìåíòàëüíàÿ ãðóïïà 3-ìíîãîîáðàçèÿ, n-ëèñòíî ðàçâåòâëåííî öèêëè åñêè íàêðûâàþùåãî (2, 1)-óçåë. Ïðè r = p ïîëó àåì ãðóïïû Ìàêëà ëàíà F p (2, 2n).

5 Êîãäà ãðóïïà G n (w 1,..., w m ) êîíå íà?

5 Êîãäà ãðóïïà G n (w 1,..., w m ) êîíå íà? 6. Êîãäà ãðóïïà G n (w 1,..., w m ) ÿâëÿåòñÿ ãðóïïîé 3-ìíîãîîáðàçèÿ?

5 Êîãäà ãðóïïà G n (w 1,..., w m ) êîíå íà? 6. Êîãäà ãðóïïà G n (w 1,..., w m ) ÿâëÿåòñÿ ãðóïïîé 3-ìíîãîîáðàçèÿ? 7. Îïèñàòü òîïîëîãè åñêîå äåéñòâèå àâòîìîðôèçìà θ n.