Regiane Lopes de Oliveira

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❘❡❧❛çã♦ ❞❡ ❛❧❝❛♥❝❡ ❡♠ ❞í❣r❛❢♦s

tr❛♥s✐t✐✈♦s ❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❩

♣♦r

❘❡❣✐❛♥❡ ▲♦♣❡s ❞❡ ❖❧✐✈❡✐r❛

❇r❛sí❧✐❛

Ficha catalográfica elaborada automaticamente, com os dados fornecidos pelo(a) autor(a)

dOL48r

de Oliveira, Regiane Lopes

Relação de alcance em dígrafos transitivos e a propriedade Z / Regiane Lopes de Oliveira;

orientador Daniela Amorim Amato. -- Brasília, 2016. 70 p.

Dissertação (Mestrado - Mestrado em Matemática) --Universidade de Brasília, 2016.

1. Dígrafos Transitivos. 2. Relações de alcance. 3. Propriedade Z. 4. Grupos Nilpotentes. I. Amato,

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❛❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t♦❞❛s ❛s ❜❡♥çã♦s ❛❧❝❛♥ç❛❞❛s✳

❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ♠✐♥❤❛ ♠ã❡✱ ♠✐♥❤❛s ✐r♠ãs ❡ ♠❡✉ ♣❛✐✳ ❊♠ ❡s♣❡❝✐❛❧ ♠✐♥❤❛ ♠ã❡ ♣❡❧♦ ❝❛r✐♥❤♦✱ ♣❡❧♦ ❛♠♦r ❡ ♣♦r ♠❡ ❛❥✉❞❛r ❛ ♥✉♥❝❛ ❞❡s✐st✐r ❞♦s ♠❡✉s s♦♥❤♦s✳

➚ ♣r♦❢❡ss♦r❛ ❉❛♥✐❡❧❛ ❆♠❛t♦ ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣❡❧❛ ♦r✐❡♥t❛çã♦✱ ♣♦r t♦❞❛ ❛ ❞❡❞✐❝❛çã♦✱ ♣❛❝✐ê♥❝✐❛✱ ❞✐s♣♦s✐çã♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❞❡♣♦s✐t❛❞❛ ❡♠ ♠✐♠ ❡ ♣♦r ♠❡ ❢❛③❡r ❡♥❝❛♥t❛r ❝♦♠ ❛ ❚❡♦r✐❛ ❞♦s ●r❛❢♦s✳ ▼✉✐t♦ ♦❜r✐❣❛❞❛ ♣♦r t✉❞♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ❈r✐st✐♥❛ ❆❝❝✐❛rr✐ ❡ ❈s❛❜❛ ❙❝❤♥❡✐❞❡r✳

❆♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❯♥❇ ♣❡❧♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡♠át✐❝♦s tr❛♥s♠✐t✐❞♦s✳ ❊♠ ❡s♣❡❝✐❛❧ às ♣r♦❢❡ss♦r❛s ❈át✐❛ ❘❡❣✐♥❛✱ ❈r✐st✐♥❛ ❆❝❝✐❛rr✐ ❡ ❉❛♥✐❡❧❛ ❆♠❛t♦✱ ♣♦r ♠❡ ❢❛③❡r❡♠ r❡❞❡s❝♦❜r✐r ♦ q✉❛♥t♦ é ♠❛r❛✈✐❧❤♦s♦ ❡st✉❞❛r ♠❛t❡♠át✐❝❛✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❛ ❝❤❡❣❛r ❛té ❛q✉✐✳ ❊s♣❡❝✐❛❧♠❡♥t❡ ♦s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❯❋▼❚✴❈❯❘✳

➚ ❙✉♥❛♠✐t❛✱ ♥ã♦ ❤á ♣❛❧❛✈r❛s ♣❛r❛ ❞❡s❝r❡✈❡r ❡ ♥❡♠ ♥ú♠❡r♦s q✉❡ ♣♦ss❛♠ ❡①♣r❡ss❛r ♦ q✉❛♥t♦ ❡✉ s♦✉ ❣r❛t❛ ♣❡❧❛ s✉❛ ❛♠✐③❛❞❡ ❡ ♣❡❧♦ s❡✉ ❛♣♦✐♦ ❞❡❞✐❝❛❞♦ ❛ ♠✐♠✳

❆ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❆♥❛✱ ❇r✉♥♦✱ ❈❤r✐st❡✱ ●❧á✉❝✐❛✱ ❑❛r❡♥✱ ❑❛r♦❧✱ ❑❡✐❞♥❛✱ ▲✉♠❡♥❛✱ ▼❛r✐❛✱ ▼❛②r❛✱ ◆❛t❤á❧✐❛✱ ❘❛❢❛❡❧✱ ❙❛r❛✱ ❲❡❧❜❡r ❡ ❲❡❧✐♥t♦♥✳ ❆❣r❛❞❡ç♦ ❛ ✈♦❝ês ♣❡❧❛ ❛♠✐③❛❞❡✱ ♣❡❧♦ ❛♣♦✐♦✱ ❡ ❝❧❛r♦✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❢❡❧✐③❡s r❡♣❧❡t♦s ❞❡ ♠✉✐t❛s r✐s❛❞❛s✳

➚s ♠✐♥❤❛s ❛♠✐❣❛s ❞❡ ♠♦r❛❞✐❛ ❆❧✐♥❡ ❡ ▲✉❝✐❛♥❛✱ ♣❡❧❛ ❝♦♥✈✐✈ê♥❝✐❛ ❢❛♥tást✐❝❛ ❡ ♣❡❧❛ ❛♠✐③❛❞❡✳ ❆♦ ❈◆P◗ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❛ ❡st❡ tr❛❜❛❧❤♦✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛♠♦s ✉♠❛ ❢❛♠í❧✐❛ ❞❡ r❡❧❛çõ❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❞❡✜♥✐❞❛s ♥♦s ✈ért✐❝❡s ❞❡ ✉♠ ❞í❣r❛❢♦✱ ❛s ❝❤❛♠❛❞❛s r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ❛♣r❡s❡♥t❛♠♦s ❞✐✈❡rs❛s ♣r♦♣r✐❡❞❛✲ ❞❡s ❣❡r❛✐s ❞❛s r❡❧❛çõ❡s ❡♠ q✉❡stã♦ ❡✱ ❡♥tã♦ ❡st✉❞❛♠♦s t❛✐s r❡❧❛çõ❡s ❡♠ ❝♦♥❡①ã♦ ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❣r✉♣♦s ❞❡ ❛✉t♦♠♦r✜s♠♦s ❞❡ ❞í❣r❛❢♦s tr❛♥s✐t✐✈♦s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❛♣r❡✲ s❡♥t❛❞♦ ♠♦str❛ q✉❡ s❡ ✉♠ ❞í❣r❛❢♦ tr❛♥s✐t✐✈♦ D ❛❞♠✐t❡ ✉♠ s✉❜❣r✉♣♦ ♥✐❧♣♦t❡♥t❡ H ❞♦ ❣r✉♣♦ ❞❡

❛✉t♦♠♦r✜s♠♦Aut(D)❞❡D✱ ❛❣✐♥❞♦ ❝♦♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ór❜✐t❛s s♦❜r❡D✱ ❡♥tã♦ ❛ ❝❧❛ss❡

❞❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡ H ❡ ♦ ♥ú♠❡r♦ ❞❡ ór❜✐t❛s ❡stã♦ ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❞❡t❡r♠✐♥❛❞❛s

♣r♦♣r✐❡❞❛❞❡s ❞❛s r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡✳

❆❧é♠ ❞✐ss♦✱ ❡st✉❞❛♠♦s ❝♦♠♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ❡stã♦ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❞í❣r❛❢♦✱ t❛✐s ❝♦♠♦ ❵t❡r ❛ ♣r♦♣r✐❡❞❛❞❡ ❩✬ ❡ ❝♦♥❞✐çõ❡s ❞❡ ❝r❡s❝✐♠❡♥t♦✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ st✉❞② ❛ ❢❛♠✐❧② ♦❢ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ❞❡✜♥❡❞ ✐♥ t❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❞✐❣r❛♣❤✱ t❤❡ ❝❛❧❧❡❞ r❡❛❝❤❛❜✐❧✐t② r❡❧❛t✐♦♥s✳ ❋✐rst✱ ✇❡ ♣r❡s❡♥t s❡✈❡r❛❧ ❣❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡❧❛t✐♦♥s ✐♥ q✉❡st✐♦♥ ❛♥❞ t❤❡♥ ✇❡ st✉❞② s✉❝❤ r❡❧❛t✐♦♥s ✐♥ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ♣r♦♣❡rt✐❡s ♦❢ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ tr❛♥s✐t✐✈❡ ❞✐❣r❛♣❤s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♠❛✐♥ r❡s✉❧t s❤♦✇s t❤❛t ✐❢ ❛ tr❛♥s✐t✐✈❡ ❞✐❣r❛♣❤D❛❞♠✐ts

❛ ♥✐❧♣♦t❡♥t s✉❜❣r♦✉♣H♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣Aut(D)♦❢D❛❝t✐♥❣ ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② ♦r❜✐ts

♦♥ D✱ t❤❡♥ t❤❡ ♥✐❧♣♦t❡♥t ❝❧❛ss ♦❢ H ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ♦r❜✐ts ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ ❞❡t❡r♠✐♥❡❞

♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡❛❝❤❛❜✐❧✐t② r❡❧❛t✐♦♥s✳

❋✉t❤❡r♠♦r❡✱ ✇❡ st✉❞② ❤♦✇ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡❛❝❤❛❜✐❧✐t② r❡❧❛t✐♦♥ ❛r❡ r❡❧❛t❡❞ ✇✐t❤ ♦t❤❡rs ♣r♦♣❡rt✐❡s ♦❢ ❞✐❣r❛♣❤s✱ s✉❝❤ ❛s ❵❤❛✈✐♥❣ ♣r♦♣❡t② ❩✬ ❛♥❞ ❣r♦✇t❤ ❝♦♥❞✐t✐♦♥s✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✐

✶ Pr❡❧✐♠✐♥❛r❡s ✶

✷ ❉í❣r❛❢♦s ❡ ●r✉♣♦s ✼

✷✳✶ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✷ Pr♦♣r✐❡❞❛❞❡ ❩ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✸ ❉í❣r❛❢♦s ❞❡ ❈❛②❧❡② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✹ ❈r❡s❝✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸ ❘❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ❡♠ ❞í❣r❛❢♦s ✶✾

✸✳✶ ❆❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡R+_k ❡ R−_k ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✸✳✶✳✶ ❆s s❡q✉ê♥❝✐❛s (R+_k)_k∈Z+ ❡ (R−

k)k∈Z+ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸✳✶✳✷ ❆s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s ❞❡R+_k ❡ R−_k ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✸✳✷ Pr♦♣r✐❡❞❛❞❡ ❩ ❡ ❝r❡s❝✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✹ ❘❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ❡ ❣r✉♣♦s ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧ ✹✶

✹✳✶ ❘❡s✉❧t❛❞♦s ❛✉①✐❧✐❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✹✳✷ ❆s r❡❧❛çõ❡sR+ _{❡ R}− _{❡♠ ❞í❣r❛❢♦s tr❛♥s✐t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼}

✹✳✸ ❖❜s❡r✈❛çõ❡s ✜♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

■♥tr♦❞✉çã♦

❯♠ ❞í❣r❛❢♦ D é ✉♠ ♣❛r (V D, ED)✱ ♦♥❞❡ V D é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡ ED ⊆V D×V D

é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ ❡❧❡♠❡♥t♦s ❞❡ V D✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ V Dsã♦ ❞❡♥♦♠✐♥❛❞♦s

✈ért✐❝❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ED ❞❡ ❛r❡st❛s✳ ❯♠ ♣❛ss❡✐♦ W = (v0, ǫ1, v1, ǫ2, . . . , ǫn, vn) ❞❡ v0 ❛té

vn é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ n+ 1 ✈ért✐❝❡s ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♣❛r❡s ❞✐st✐♥t♦s✮ v0✱ v1✱ . . .✱ vn ❡ n

✐♥❞✐❝❛❞♦r❡s ǫ1, . . . , ǫn∈ {−1,1}✱ t❛❧ q✉❡ ♣❛r❛ t♦❞♦j∈ {1,2, . . . , n}t❡♠♦s✿

ǫj = 1⇒(vj−1, vj)∈EW

ǫj =−1⇒(vj, vj−1)∈EW

❯♠ ♣❛ss❡✐♦ é ❞✐t♦ ❛❧t❡r♥❛❞♦ s❡ ♦s ✐♥❞✐❝❛❞♦r❡s ǫi ❛❧t❡r♥❛♠✳

❊♠ ❬✺❪ ❢♦✐ ✐♥tr♦❞✉③✐❞❛ ❛ ♥♦çã♦ ❞❡ r❡❧❛çã♦ ❞❡ ❛❧❝❛♥❝❡ ❞❡✜♥✐❞❛ ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ❞❡ ✉♠ ❞í❣r❛❢♦✱ ❞❡♥♦t❛❞❛ ♣♦r A✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❛r❡st❛e′ é ❛❧❝❛♥ç❛❞❛ ♣♦r ✉♠❛ ❛r❡st❛e✱ s❡ ❡①✐st❡ ✉♠

♣❛ss❡✐♦ ❛❧t❡r♥❛❞♦ ❝✉❥❛ ❛ ♣r✐♠❡✐r❛ ❛r❡st❛ é e❡ ❛ ú❧t✐♠❛ ❛r❡st❛ é e′✳

❊♠ ❬✺❪ ♦s ❛✉t♦r❡s t❡♥t❛r❛♠ ❝❧❛ss✐✜❝❛r ❞í❣r❛❢♦s ❛❧t❛♠❡♥t❡ tr❛♥s✐t✐✈♦s ✉s❛♥❞♦ ♣r♦♣r✐❡❞❛❞❡s ❞❛ r❡❧❛çã♦ A✱ ❝♦♠ ✐st♦ ❢♦r❛♠ ♣r♦♣♦st♦ ✉♠ ♥ú♠❡r♦ ✐♥t❡r❡ss❛♥t❡ ❞❡ ♣r♦❜❧❡♠❛s✳ ❆❧❣✉♥s ❞❡st❡s

♣r♦❜❧❡♠❛s ❢♦r❛♠ ❡st✉❞❛❞♦s✱ ❡♠ ❬✶✵❪ ♣♦r ▼❛❧♥✐cˇ✱ ▼❛r✉sˇ✐ˇc✱ ❙❡✐❢t❡r ❡ ❩❣r❛❜❧✐cˇ❡ ❝♦♠♦ ❢r✉t♦ ❞❡st❡ ❡st✉❞♦ s✉r❣✐✉ ♦ ✐♥t❡r❡ss❡ ❞❡ ❡st✉❞❛r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ❛❣♦r❛ s❡♥❞♦ ❞❡✜♥✐❞❛ ♥♦s ✈ért✐❝❡s ❞❡ ✉♠ ❞í❣r❛❢♦✳ ❆ss✐♠✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠_Sˇ_{♣❛r❧✱ ❞❡✜♥✐r❛♠ ❡♠ ❬✶✶❪ r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ♥♦s} ✈ért✐❝❡s✳

❯♠❛ ❞❡✜♥✐çã♦ ✐♥❢♦r♠❛❧ ✭♣❛r❛ ✉♠❛ ❞❡✜♥✐çã♦ ♣r❡❝✐s❛ ✈❡❥❛ s❡çã♦ ✸✮ é ❛ s❡❣✉✐♥t❡✳ Pr✐♠❡✐r♦✱ ♦ ♣❡s♦ ❞❡ ✉♠ ♣❛ss❡✐♦ W é ❞❡✜♥✐❞♦ ❝♦♠♦ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ♣❡r❝♦rr✐❞❛s ♥♦

♠❡s♠♦ s❡♥t✐❞♦ ❞❡ W✱ ❡ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ♣❡r❝♦rr✐❞❛s ♥♦ s❡♥t✐❞♦ ❝♦♥trár✐♦ ❛ W✳ ❙❡❥❛ k≥ 1 ✉♠ ✐♥t❡✐r♦✳ ❊♥tã♦ ✉♠ ✈ért✐❝❡ u ❡stá R+_k r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ✉♠ ✈ért✐❝❡ v✱ s❡ ❡①✐st❡ ✉♠ ♣❛ss❡✐♦W

❞❡ ♣❡s♦ 0 ❞❡ u ❛ v t❛❧ q✉❡ t♦❞♦ s✉❜♣❛ss❡✐♦ ❞❡ W ❝♦♠ ✈ért✐❝❡ ✐♥✐❝✐❛❧ u t❡♠ ♣❡s♦ ✈❛r✐❛♥❞♦ ♥♦

✐♥t❡r✈❛❧♦ [0, k]✳ ❉❡ ❢♦r♠❛ s✐♠✐❧❛r✱ ❞✐③❡♠♦s q✉❡ ✉♠ ✈ért✐❝❡u ❡stáR−_k r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ✉♠ ✈ért✐❝❡ v✱ s❡ ❡①✐st❡ ✉♠ ♣❛ss❡✐♦ W ❞❡ ♣❡s♦ 0 ❧✐❣❛♥❞♦ u ❛ v t❛❧ q✉❡ t♦❞♦ s✉❜♣❛ss❡✐♦ ❞❡ W ❝♦♠ ✈ért✐❝❡

■♥tr♦❞✉çã♦ ✐✐

♣❛r❛ ❛❧❣✉♠ k ❡♥tã♦ ❞❡✜♥✐♠♦s ♦ ❡①♣♦❡♥t❡exp+(D) ❞❡ ✉♠ ❞í❣r❛❢♦ D✱ ❝♦♠♦ ♦ ♠❡♥♦r ✐♥t❡✐r♦ t❛❧

q✉❡ R+_k =R+_k+1✳ ❆❣♦r❛ s❡ R+_k =6 R+_k+1✱ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k✱ ❡♥tã♦exp+(D) =∞✳ ❙✐♠✐❧❛r♠❡♥t❡ ❞❡✜♥✐♠♦s ♦ ❡①♣♦❡♥t❡ exp−_(D)✳

❚❛♠❜é♠ ❡♠ ❬✺❪ ❢♦✐ ♣r♦♣♦st♦ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✿

Pr♦❜❧❡♠❛ ✶✳ ❊①✐st❡ ✉♠ ❞í❣r❛❢♦ ❝♦♥❡①♦✱ ❛❧t❛♠❡♥t❡ ❛r❝♦ tr❛♥s✐t✐✈♦ ❡ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ t❛❧ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❛❧❝❛♥❝❡ ♥❛s ❛r❡st❛s é ✉♥✐✈❡rs❛❧❄

❊st❡ ♣r♦❜❧❡♠❛ ✜❝♦✉ ✉♠ ❜♦♠ t❡♠♣♦ ❡♠ ❛❜❡rt♦✱ s❡♥❞♦ ✜♥❛❧♠❡♥t❡ s♦❧✉❝✐♦♥❛❞♦ ❡♠ ❬✻❪✳ ❱❡❥❛ q✉❡ ♦ Pr♦❜❧❡♠❛ ✶ t❛♠❜é♠ é s♦❧✉❝✐♦♥❛❞♦✱ ✉s❛♥❞♦ r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ♥♦s ✈ért✐❝❡s✱ s❡ s✉♣♦r♠♦s ❛♣❡♥❛s q✉❡ ♦ ❞í❣r❛❢♦ é ❝♦♥❡①♦ ❡ t❡♠ ✉♠ ❧❛ç♦ ❡♠ ❝❛❞❛ ✈ért✐❝❡✱ ♦✉ s❡ ♦ ❞í❣r❛❢♦ é ❝♦♥❡①♦✱ t❡♥❞♦ ✉♠ ❝✐❝❧♦ ❞✐r❡❝✐♦♥❛❞♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦2❡♠ ❝❛❞❛ ✈ért✐❝❡ ❡ ❝♦♥té♠ ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ í♠♣❛r ✭✈❡❥❛ Pr♦♣♦s✐çã♦ ✸✳✻✮✳

❚❡♠♦s t❛♠❜é♠ q✉❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ R_k+ ❡ R_k− ❡stã♦ ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♣r♦♣r✐❡✲

❞❛❞❡ ❞❡ ❞í❣r❛❢♦s✱ t❛❧ ❝♦♠♦ ❝♦♥❞✐çã♦ ♣❛r❛ ✉♠ ❞í❣r❛❢♦ t❡r ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❝♦♠♦ ♣♦❞❡ s❡r ✈✐st♦ ♥♦ t❡♦r❡♠❛ ❛ s❡❣✉✐r✳

❚❡♦r❡♠❛ ✸✳✷✶✳ ❙❡❥❛ D ✉♠ ❞í❣r❛❢♦ ❝♦♥❡①♦✱ tr❛♥s✐t✐✈♦ ❡ ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✳ ❙❡ ♣❡❧♦ ♠❡♥♦s ✉♠

❞♦s ❡①♣♦❡♥t❡s exp+_{(D) ❡ exp}−_{(D) é ✐♥✜♥✐t♦ ❡♥tã♦Dt❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✳}

❈♦♠ ✐st♦ t❡♠♦s q✉❡ ✉♠ ❣r✉♣♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ s❡ ♣❛r❛ ✉♠ ❞♦s ❞í❣r❛❢♦s ❞❡ ❈❛②❧❡② D ❞❡G✱ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❡①♣♦❡♥t❡s exp+(D) ❡exp−(D)é ✐♥✜♥✐t♦✳

❉❡ss❛ ❢♦r♠❛✱ s❡G♥ã♦ t❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✱ ❛♠❜♦s ♦s ❡①♣♦❡♥t❡sexp+_(D)❡exp−_(D)

sã♦ ✜♥✐t♦s✳ ◆❡st❡ ❝❛s♦✱ s✉r❣❡ ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛✿

P❡r❣✉♥t❛✳ ❖ q✉❡ ♣♦❞❡ s❡r ❞✐t♦ s♦❜r❡ ♣r♦♣r✐❡❞❛❞❡s ❞❛s r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ❡♠ ❞í❣r❛❢♦s ❞❡ ❈❛②❧❡② ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧❄

P♦r ✉♠ r❡s✉❧t❛❞♦ ❞❡ ●r♦♠♦✈✱ ❡♠ ❬✽❪✱ t❡♠♦s q✉❡ ✉♠ ❣r✉♣♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ t❡♠ ❝r❡s❝✐✲ ♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❡ ❝♦♥té♠ ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ♥✐❧♣♦t❡♥t❡ ❝♦♠ í♥❞✐❝❡ ✜♥✐t♦✳ ❆ss✐♠✱ ❛ r❡s♣♦st❛ ♣❛r❛ ❛ ♣❡r❣✉♥t❛ ❛❝✐♠❛ é ❞❛❞❛ ♣❡❧♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿

❚❡♦r❡♠❛ ✹✳✶✺✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❛❣✐♥❞♦ tr❛♥s✐t✐✈❛♠❡♥t❡ s♦❜r❡ ✉♠ ❞í❣r❛❢♦ ❝♦♥❡①♦ D ❡ s❡❥❛ N EG ♥♦r♠❛❧ ♥✐❧♣♦t❡♥t❡ ❞❡ ❝❧❛ss❡ r ❛❣✐♥❞♦ ❝♦♠ m ór❜✐t❛s s♦❜r❡ D✱ ♦♥❞❡ 1≤m < ∞✳ ❊♥tã♦

exp+(D) =exp−(D)≤m(r+ 1)−1✳

❚❡♠♦s t❛♠❜é♠ q✉❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ R_k+ ❡ R_k− ❡stã♦ ✐♥t✐♠❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♣r♦♣r✐❡✲

❞❛❞❡s ❞❡ ❞í❣r❛❢♦s✱ t❛❧ ❝♦♠♦ ❝♦♥❞✐çã♦ ♣❛r❛ t❡r ♣r♦♣r✐❡❞❛❞❡ Z ✭✈❡❥❛ Pr♦♣♦s✐çõ❡s✸✳✷✵ ❡✸✳✶✾✮✳ ❯♠

■♥tr♦❞✉çã♦ ✐✐✐

❞í❣r❛❢♦ q✉❡ t❡♠ ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s Z❡ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s {(i, i+ 1)|i∈Z}✳

❉✐❛♥t❡ ❞✐ss♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❡st✉❞❛r ❛s r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡ ❞❡✜♥✐❞❛s ♥♦s ✈ért✐❝❡s ❞❡ ✉♠ ❞í❣r❛❢♦✱ t❛✐s ❝♦♠♦ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ s✉❛s ❝♦♥❡①õ❡s ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❣r✉♣♦ ❞❡ ❛✉t♦♠♦r✜s♠♦ ❞❡ ❞í❣r❛❢♦s tr❛♥s✐t✐✈♦s ❡ ♣r♦♣r✐❡❞❛❞❡ Z✳

❚❛❧ ❡st✉❞♦ s❡rá r❡❛❧✐③❛❞♦ ❛tr❛✈és ❞❡ r❡s✉❧t❛❞♦s ❞❡♠♦♥str❛❞♦s ❡♠ ❬✶✶❪ ❡ ❬✶✸❪ ❡ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✶ r❡❧❡♠❜r❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s ✉t✐✲ ❧✐③❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳ ◆♦ ❝❛♣ít✉❧♦ ✷ ❞❛r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ❞❛ t❡♦r✐❛ ❞❡ ❞í❣r❛❢♦s✳ ◆♦ ❝❛♣ít✉❧♦ ✸ ❡st✉❞❛r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ❞❡ R+_k ❡ R−_k ❡ s✉❛s ❝♦♥❡①õ❡s ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❞í✲

❣r❛❢♦s t❛✐s ❝♦♠♦ ♣r♦♣r✐❡❞❛❞❡ Z ❡ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✱ ♥❛ q✉❛❧ t❛♠❜é♠ s❡rá ❛♣r❡s❡♥t❛❞♦ ♦

❚❡♦r❡♠❛ ✸✳✷✶ ❡①✐❜✐❞♦ ❛❝✐♠❛✳ ◆♦ ❝❛♣ít✉❧♦ ✹❡st✉❞❛r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❞í❣r❛❢♦s ❞❡ ❈❛②❧❡② ❞❡ ❣r✉♣♦s ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦s ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧ ♦❜t✐❞❛s ❛tr❛✈és ❞❛s r❡❧❛çõ❡s ❞❡ ❛❧❝❛♥❝❡✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛♣r❡s❡♥t❛♠♦s ♦ ❚❡♦r❡♠❛ ✹✳✶✺ ❡①✐❜✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ q✉❡ é ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❋✐♥❛❧✐③❛r❡♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠❛ ❝♦♥❥❡❝t✉r❛✱ ❛ q✉❛❧ s❡ ❢♦r ✈❡r❞❛❞❡✐r❛ ❢♦r♥❡❝❡ ✉♠❛ r❡s♣♦st❛ ♣♦s✐t✐✈❛ ♣❛r❛ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ●r✐❣♦r❝❤✉❝❦✿

❈❛♣ít✉❧♦

1

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ ❞❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❞❛ t❡♦r✐❛ ❞❡ ❣r✉♣♦s✱ q✉❡ s❡rã♦ ✉s❛❞♦s ❛♦ ❧♦♥❣♦ ❞♦ tr❛❜❛❧❤♦✳ ❆❧❣✉♥s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ❛q✉✐ ❥á ❢❛③❡♠ ♣❛rt❡ ❞❡ q✉❛❧q✉❡r ❝✉rs♦ ❜ás✐❝♦ ❞❡ ❣r✉♣♦s✱ s❡♥❞♦ ❛ss✐♠ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s s❡rã♦ ♦♠✐t✐❞❛s✳ P❛r❛ ✉♠ ♠❛✐♦r ❛♣r♦❢✉♥❞❛♠❡♥t♦ ♥♦ ❛ss✉♥t♦✱ ✐♥❞✐❝❛♠♦s ❬✾❪✱ ❬✼❪ ❡ ❬✶✼❪✳

❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ s❡❥❛ Ω ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦✳ ❙✉♣♦♥❤❛ q✉❡ ♣❛r❛ ❝❛❞❛ g ∈ G ❡ ω ∈ Ω ❞❡✜♥✐♠♦s ✉♠ ❡❧❡♠❡♥t♦ g_{ω∈Ω✳ ❉✐③❡♠♦s q✉❡ ✐st♦ ❞❡✜♥❡ ✉♠❛ ❛çã♦ ❞❡Gs♦❜r❡Ω✭♦✉G❛❣❡ s♦❜r❡}

Ω ♦✉ q✉❡ Ωé ✉♠ G✲❡s♣❛ç♦✮✱ s❡ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

✐✮ 1_{ω=ω✱ ♣❛r❛ t♦❞♦ ω∈Ω✱ ♦♥❞❡ 1 ❞❡♥♦t❛ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ G✳}

✐✐✮ g₍h_{ω) =} gh_{ω✱ ♣❛r❛ t♦❞♦ ω∈Ω ❡ t♦❞♦g, h∈G✳}

❊①❡♠♣❧♦ ✶✳✶✳ ❖ ❣r✉♣♦ s✐♠étr✐❝♦ G =Sym(Ω) ✭é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❜✐❥❡çõ❡s ❞❡ Ω ❡♠ Ω ♠✉♥✐❞♦ ❞❛ ♦♣❡r❛çã♦ ❝♦♠♣♦s✐çã♦ ◦ ✮ t❡♠ ✉♠❛ ❛çã♦ ♥❛t✉r❛❧ s♦❜r❡ Ω✱ ♣♦✐s ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ G é

✉♠❛ ♣❡r♠✉t❛çã♦ ❞❡ Ω✱ ♦✉ s❡❥❛✱ ♦ ❡❧❡♠❡♥t♦ g_{ω:=g(ω) é ❛ ✐♠❛❣❡♠ ♣♦rg ❞❡ ω✳ ❆ss✐♠}

✐✮ e_{ω=e(ω) =ω✱ ♦♥❞❡ eé ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ Sym(Ω)✳}

✐✐✮ g₍h_{ω) =} g_{(h(ω)) =g(h(ω)) =g◦h(ω) =} gh_ω✳

❊①❡♠♣❧♦ ✶✳✷✳ ❈♦♥s✐❞❡r❡ Ω =G ❡ ♣❛r❛ ❝❛❞❛x∈G❞❡✜♥❛ ✉♠ ❡❧❡♠❡♥t♦ g_{x:=gx✱ ♦♥❞❡ g∈G✳}

❊♥tã♦ t❡♠♦s q✉❡

✐✮ 1_{x=x✱ ♦♥❞❡ ✶ é ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ G✳}

✐✐✮ gh_{x= (gh)x=g(hx) =g(}h_{x) =} g₍h_{x)✱ ♣❛r❛ t♦❞♦ g, h∈G✳}

▲♦❣♦✱ G❛❣❡ s♦❜r❡ Ω✳ ❚❛❧ ❛çã♦ é ❞✐t❛ ✉♠❛ ❛çã♦ ♣♦r ♠✉❧t✐♣❧✐❝❛çã♦ à ❡sq✉❡r❞❛ ✭♦✉ q✉❡ G❛❣❡ ♣♦r

✷

❙❡❥❛♠G ✉♠ ❣r✉♣♦✱Ω✉♠G✲❡s♣❛ç♦ ❡ω ∈Ω✳ ❉❡✜♥✐♠♦s ❛ ór❜✐t❛ ❞❡ ω✱ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦

G_ω:={ g_ω|g∈G}.

❊ ♦ ❡st❛❜✐❧✐③❛❞♦r ❞❡ω ❡♠G✱ é ♦ ❝♦♥❥✉♥t♦

Gw :={g∈G|gω=ω}.

➱ ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡Gω é s✉❜❣r✉♣♦ ❞❡G✳

❉✐③❡♠♦s q✉❡ ❛ ❛çã♦ ❞❡G s♦❜r❡ Ω é tr❛♥s✐t✐✈❛ ✭♦✉ q✉❡Ω é ✉♠G✲❡s♣❛ç♦ tr❛♥s✐t✐✈♦✮ s❡ ♣❛r❛

t♦❞♦ x, y∈Ω❡①✐st❡g∈Gt❛❧ q✉❡ g_{x=y✱ ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ s❡G❛❣❡ ❝♦♠ ✉♠❛ ú♥✐❝❛ ór❜✐t❛}

s♦❜r❡ Ω✳ ❙❡ ❛❧é♠ ❞✐ss♦✱ t❛❧ ❡❧❡♠❡♥t♦ é ú♥✐❝♦✱ ❡♥tã♦ ❛ ❛çã♦ é ❞✐t❛ r❡❣✉❧❛r ✭♦✉ G❛❣❡ r❡❣✉❧❛r♠❡♥t❡

s♦❜r❡ Ω✮✳ ▲♦❣♦✱ ❛ ❛çã♦ é r❡❣✉❧❛r s❡ é tr❛♥s✐t✐✈❛ ❡Gx={1} ♣❛r❛ t♦❞♦x∈Ω✳

❆ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❛❜❛✐①♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬❬✼❪✱ ❚❡♦r❡♠❛ ✶✳✹❆❪✳

❚❡♦r❡♠❛ ✶✳✸✳ ❙✉♣♦♥❤❛ q✉❡ G ❛❣❡ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ Ω❡ q✉❡ x, y∈Ω ❡ g∈G✳ ❊♥tã♦

✐✮ ❉✉❛s ór❜✐t❛s G_{x ❡} G_{y sã♦ ✐❣✉❛✐s ✭❝♦♠♦ ❝♦♥❥✉♥t♦s✮ ♦✉ sã♦ ❞✐s❥✉♥t❛s✱ ❧♦❣♦ ♦ ❝♦♥❥✉♥t♦ ❞❡}

t♦❞❛s ❛s ór❜✐t❛s é ✉♠❛ ♣❛rt✐çã♦ ❞❡ Ω✳

✐✐✮ ❖ ❡st❛❜✐❧✐③❛❞♦rGx =gGyg−1 s❡♠♣r❡ q✉❡x= g_{y✱ ♣❛r❛ t♦❞♦ x, y∈Ω ❡ g∈G✳}

✐✐✐✮ |G_{x|=|G:Gx|,♦♥❞❡xé ✉♠ ❡❧❡♠❡♥t♦ q✉❛❧q✉❡r ❞❡Ω✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡G❛❣❡ r❡❣✉❧❛r♠❡♥t❡}

s♦❜r❡ Ω ❡♥tã♦|G|=|Ω|✳

❖❜s❡r✈❛çã♦ ✶✳✹✳ ❙❡ G ≤ Aut(Ω) ✭❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ❛✉t♦♠♦r✜s♠♦ ❞❡ Ω ❡♠ Ω ♠✉♥✐❞♦ ❞❛ ♦♣❡r❛çã♦ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s✮ é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❛❣✐♥❞♦ tr❛♥s✐t✐✈❛♠❡♥t❡ s♦❜r❡ Ω ❡♥tã♦ G

❛❣❡ r❡❣✉❧❛r♠❡♥t❡ s♦❜r❡ Ω✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ G ❛❣❡ tr❛♥s✐t✐✈❛♠❡♥t❡✱ ❡♥tã♦ r❡st❛ ♣r♦✈❛r♠♦s q✉❡ ♦

❡st❛❜✐❧✐③❛❞♦r Gx ={1}✱ ♣❛r❛ t♦❞♦ x ∈ Ω✳ ❈♦♠♦ G é ❛❜❡❧✐❛♥♦✱ t❡♠♦s ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✸ q✉❡ ♦s

❡st❛❜✐❧✐③❛❞♦r❡s ❡❧❡♠❡♥t♦s ❞❡ Ω❝♦✐♥❝✐❞❡♠✳ ❉❡st❡ ♠♦❞♦ s❡g∈G✜①❛ ❛❧❣✉♠ ❡❧❡♠❡♥t♦ ❞❡Ω✱ ❡♥tã♦

g ✜①❛ t♦❞♦s ❡❧❡♠❡♥t♦s ❞❡ Ω✳ ❈♦♠♦ ♦ ú♥✐❝♦ ❛✉t♦♠♦r✜s♠♦ q✉❡ ✜①❛ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ Ω é ❛ ✐❞❡♥t✐❞❛❞❡✱ s❡❣✉❡ q✉❡ Gx={1}✱ ♣❛r❛ t♦❞♦ x∈Ω✳

❈♦♠♦Ωé ✉♠ G✲❡s♣❛ç♦✱ ♣❛r❛ ❝❛❞❛g∈G✱ ❞❡✜♥❛ ✉♠❛ ❛♣❧✐❝❛çã♦

ρg : Ω → Ω

ω → gω.

◆♦t❡ q✉❡ t❛❧ ❛♣❧✐❝❛çã♦ ❞❡✜♥❡ ✉♠❛ ❛çã♦ ❞❡Gs♦❜r❡Ω✳ ❆❣♦r❛✱ ♣r♦✈❛r❡♠♦s q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❛❝✐♠❛ é ✉♠❛ ❜✐❥❡çã♦✱ ♠♦str❛♥❞♦ q✉❡ ❝❛❞❛ ρg t❡♠ ✉♠ ✐♥✈❡rs♦✱ ♦♥❞❡ g ∈ G✳ ❙❡❥❛♠ g, h∈ G ❡ ω ∈Ω✱ ❡♥tã♦

✸

❆ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ é ♦❜t✐❞❛ ❞♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛çã♦✳ ❆ss✐♠✱ (ρg ◦ρh) = ρgh✳ ❊♠

♣❛rt✐❝✉❧❛r ♣❛r❛ h=g−1 t❡♠♦s✿

ρg◦ρg−1 =ρ_gg−1 =ρ_e=ρ_g−1_g=ρ_g−1 ◦ρ_g.

❊ ✐st♦ ✐♠♣❧✐❝❛ q✉❡ (ρg)−1 =ρg−1✱ ❝♦♠♦ q✉❡rí❛♠♦s✳ P♦rt❛♥t♦✱ ρg ∈ Sym(Ω)✳ ❙❡❣✉❡ q✉❡ t❡♠♦s

✉♠❛ ❛♣❧✐❝❛çã♦

ρ:G → Sym(Ω)

g → ρ(g) :=ρg

❆❧é♠ ❞✐ss♦✱ ρ é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳ ❉❡ ❢❛t♦✱ s❡❥❛♠ g, h∈G✱ ❡♥tã♦

ρ(gh) =ρgh=ρgρh =ρ(g)ρ(h).

❉❡♥♦t❛r❡♠♦s ♣♦r id ♦ ❡❧❡♠❡♥t♦ ✐❞❡♥t✐❞❛❞❡ ❞❡Sym(Ω)✳ ❖ ♥ú❝❧❡♦ ❞❡ ρ✱ ❞❛❞♦ ♣♦r

kerρ={g∈G|ρg =id}={g∈G|ρg(ω) =id,∀ ω∈Ω}={g∈G|g_{ω=ω,∀ ω∈Ω},}

é ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G❡ ♣❡❧♦ 1◦ _{❚❡♦r❡♠❛ ❞♦ ✐s♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✿}

G

kerρ ∼=Imρ≤Sym(Ω)

♦♥❞❡ Imρ❞❡♥♦t❛ ❛ ✐♠❛❣❡♠ ❞❡ ρ✳ ❉✐③❡♠♦s q✉❡ ρ(G) =Imρ é ♦ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çã♦ ✐♥❞✉③✐❞♦

♣♦r G ❡♠ Ω✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡kerρ ={1}✱ ❞✐③❡♠♦s q✉❡ ❛ ❛çã♦ ❞❡ G s♦❜r❡ Ω é ✜❡❧✱ ♦✉ q✉❡ G

❛❣❡ ✜❡❧♠❡♥t❡✳ ❊ ♥❡st❡ ❝❛s♦✱ ♣♦❞❡♠♦s ♣❡♥s❛r ❡♠ G❝♦♠♦ ✉♠ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çã♦ ❡♠ Ω✳ ❈❛s♦ ❝♦♥trár✐♦✱ G é ✉♠ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çã♦ ♠ó❞✉❧♦ kerρ✳ ❊ t❡♠♦s t❛♠❜é♠ q✉❡ G

kerρ ❛❣❡ ✜❡❧♠❡♥t❡

s♦❜r❡ Ω✳

❆ss✉♠❛ ❛❣♦r❛ q✉❡ ♦ ❝♦♥❥✉♥t♦Ωé ✉♠G✲❡s♣❛ç♦ tr❛♥s✐t✐✈♦✳ ❙❡❥❛≈✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛

s♦❜r❡ Ω✳ ❙❡≈é ✐♥✈❛r✐❛♥t❡ ♣❡❧❛ ❛çã♦ ❞❡ G✱ ♦✉ s❡❥❛✱

x ≈ y ⇔ g_{x ≈} g_y

♣❛r❛ t♦❞♦x, y∈Ω❡g∈G✱ ❞✐③❡♠♦s q✉❡≈é ✉♠❛G✲❝♦♥❣r✉ê♥❝✐❛ s♦❜r❡Ω✳ ❯♠❛G✲❝♦♥❣r✉ê♥❝✐❛ é

❞✐t❛ ♥ã♦ tr✐✈✐❛❧ s❡ ❡①✐st❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❝♦♠ ♠❛✐s ❞❡ ✉♠ ❡❧❡♠❡♥t♦✱ ❡ é ❞✐t❛ ♣ró♣r✐❛ s❡ ❡①✐st❡ ♠❛✐s ❞♦ q✉❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳

❊①❡♠♣❧♦ ✶✳✺✳ P❛r❛ G:=GL(2,R) ❛❣✐♥❞♦ s♦❜r❡ R2_{\ {(0,0)}✱ ❛ r❡❧❛çã♦}

✹

é ✉♠❛ G✲❝♦♥❣r✉ê♥❝✐❛✳ ❙❡ ✐❞❡♥t✐✜❝❛r♠♦s R2 _{❝♦♠ ♦ ♣❧❛♥♦✱ ❡♥tã♦ ❛s≈✲❝❧❛ss❡s sã♦ r❡t❛s ♣❛ss❛♥❞♦} ♣❡❧❛ ❛ ♦r✐❣❡♠ ♠❛s ♥ã♦ ❝♦♥t❡♥❞♦ ❛ ♦r✐❣❡♠✳

❆ r❡❧❛çã♦ ❡♥tr❡ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ ✉♠ ❣r✉♣♦ tr❛♥s✐t✐✈♦G❡ ✉♠❛G✲❝♦♥❣r✉ê♥❝✐❛ é ❞❛❞❛ ♣❡❧♦

s❡❣✉✐♥t❡ t❡♦r❡♠❛✳

❚❡♦r❡♠❛ ✶✳✻✳ ❙❡❥❛ Ω ✉♠ G✲❡s♣❛ç♦ tr❛♥s✐t✐✈♦ ❡ s❡❥❛ H ✉♠ s✉❜❣r✉♣♦ ♥♦r♠❛❧ ❞❡ G✳ ❊♥tã♦ ❛s

ór❜✐t❛s ❞❡ H sã♦ ❛s ❝❧❛ss❡s ❞❡ ✉♠❛G✲❝♦♥❣r✉ê♥❝✐❛ s♦❜r❡ Ω✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x, y∈Ω✳ ❉❡✜♥❛ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦

x≈y⇔ ❡①✐st❡ ❛❧❣✉♠h∈H t❛❧ q✉❡ hx=y.

◆♦t❡ q✉❡ ≈é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ ❛s ❝❧❛ss❡s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s sã♦ ❛s ór❜✐t❛s ❞❡H✳

❙✉♣♦♥❤❛ q✉❡ x≈y✱ ❧♦❣♦ ❡①✐st❡ h ∈H t❛❧ q✉❡ h_{x =y✳ ◗✉❡r❡♠♦s ♠♦str❛r q✉❡} g_x≈ g_{y ✱}

g ∈G✳ ❱❡❥❛ q✉❡✱ gy= ghx✳ ❈♦♠♦ HEG❡♥tã♦ ❡①✐st❡ ❛❧❣✉♠ h′ ∈H t❛❧ q✉❡ gh=h′g✳ ❆ss✐♠✱

g_y= gh_x= h′_g

x✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ g_x≈ g_{y✱ ♣♦✐sh}′ _∈H✳

❆❣♦r❛✱ s❡ g_x≈ g_{y ❡♥tã♦} g_y= hg_{x✱ ♣❛r❛ ❛❧❣✉♠h∈H✱ ❧♦❣♦y=} g−1_hg

x ❡✱ ♣♦rt❛♥t♦x≈y✱

♣♦✐sg−1_hg∈H✳

❉❡✜♥✐çã♦ ✶✳✼✳ ❙❡❥❛ Ω ✉♠G✲❡s♣❛ç♦ tr❛♥s✐t✐✈♦✳ ❙❡ ♥ã♦ ❡①✐st❡♠G✲❝♦♥❣r✉ê♥❝✐❛s ♣ró♣r✐❛s ❡ ♥ã♦✲

tr✐✈✐❛✐s ❡♥tã♦ ❞✐③❡♠♦s q✉❡ Ω é ✉♠ G✲❡s♣❛ç♦ ♣r✐♠✐t✐✈♦ ♦✉ q✉❡ G ❛❣❡ ♣r✐♠✐t✐✈❛♠❡♥t❡ s♦❜r❡ Ω✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞✐③❡♠♦s q✉❡ G❛❣❡ ✐♠♣r✐♠✐t✐✈❛♠❡♥t❡ s♦❜r❡ Ω✳

❆ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ✐♠♣❧✐❝❛ q✉❡ s❡≈é ✉♠❛G✲❝♦♥❣r✉ê♥❝✐❛ s♦❜r❡ ✉♠G✲❡s♣❛ç♦ ♣r✐♠✐t✐✈♦✱ ❡♥tã♦

≈é tr✐✈✐❛❧ ✭t♦❞❛s ❛s ❝❧❛ss❡s t❡♠ s♦♠❡♥t❡ ✉♠ ❡❧❡♠❡♥t♦✮ ♦✉ ✉♥✐✈❡rs❛❧ ✭❡①✐st❡ s♦♠❡♥t❡ ✉♠❛ ❝❧❛ss❡✱

t♦❞♦ Ω✮✳

❉❡✜♥✐çã♦ ✶✳✽✳ ❯♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ∆⊆Ωé ✉♠ ❜❧♦❝♦ s❡ ♣❛r❛ t♦❞♦g∈G✱ ♦✉∆∩ g_{∆ =∅}

♦✉ ∆ = g∆✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❜❧♦❝♦ é ♥ã♦ tr✐✈✐❛❧ s❡ |∆|>1✱ ❡ ♣ró♣r✐♦ s❡ ∆6= Ω✳

❊①❡♠♣❧♦ ✶✳✾✳ ❙❡ G❛❣❡ tr❛♥s✐t✐✈❛♠❡♥t❡ s♦❜r❡Ω❡ s❡ ∆❡ Γ sã♦ ❜❧♦❝♦s ❞❡G❝♦♥t❡♥❞♦ ✉♠ ♣♦♥t♦

✺

❉❡✜♥✐çã♦ ✶✳✶✵✳ ❙❡❥❛ G✉♠ ❣r✉♣♦ ❛❣✐♥❞♦ tr❛♥s✐t✐✈❛♠❡♥t❡ s♦❜r❡ Ω✳ ❙❡❥❛∆✉♠ ❜❧♦❝♦✳ ❖ s✐st❡♠❛ ❞❡ ❜❧♦❝♦s ❝♦♥t❡♥❞♦ ∆✱ é ♦ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣♦r

Σ :={x_{∆ |x∈G}.}

❯♠ s✐st❡♠❛ ❞❡ ❜❧♦❝♦s Σ é ❞✐t♦ s✐st❡♠❛ ❞❡ ✐♠♣r✐♠✐t✐✈✐❞❛❞❡ s❡ Σ é ❢♦r♠❛❞♦ ♣♦r ❜❧♦❝♦s ♥ã♦ tr✐✈✐❛✐s✳ ❖❜s❡r✈❡ q✉❡ ✉♠ s✐st❡♠❛ ❞❡ ❜❧♦❝♦s Σ é ✉♠❛ ♣❛rt✐çã♦ ❞❡Ω❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ Σé ✉♠ ❜❧♦❝♦✳

❉❡✜♥✐çã♦ ✶✳✶✶✳ ❙❡❥❛ ∆ ✉♠ ❜❧♦❝♦✳ ❖ ❡st❛❜✐❧✐③❛❞♦r ❞❡ ∆ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ G q✉❡

❞❡✐①❛ ∆✐♥✈❛r✐❛♥t❡✱ ✐st♦ é✱

G{∆}:={g∈G| g∆ = ∆}.

◆♦t❡ q✉❡ G_{∆} é ✉♠ s✉❜❣r✉♣♦ ❞❡G✳

▲❡♠❛ ✶✳✶✷✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❛❣✐♥❞♦ tr❛♥s✐t✐✈❛♠❡♥t❡ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ Ω ❡ s❡❥❛ ∆ ✉♠ ❜❧♦❝♦ ♣❛r❛ G✳ ❊♥tã♦ G{∆} ❛❣❡ tr❛♥s✐t✐✈❛♠❡♥t❡ s♦❜r❡ ∆✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x, y ∈∆✳ P❡❧❛ tr❛♥s✐t✐✈✐❞❛❞❡ ❞❡ G ❡①✐st❡ g∈G t❛❧ q✉❡y = g_{x✳ ❆ss✐♠✱}

y ∈∆ ❡ y = g_x∈ g_{∆✱ ✐st♦ é✱ ∆∩} g_{∆✱ ❞❡ ♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ ∆ =} g_{∆✱ ♣♦✐s ∆é ✉♠ ❜❧♦❝♦✳}

P♦rt❛♥t♦ g∈G{∆}✳

P❛r❛ ✜♥❛❧✐③❛r ❡st❡ ❝❛♣ít✉❧♦ ❞❛r❡♠♦s ❛❣♦r❛ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ❞❡ ❣r✉♣♦s ❛❜str❛t♦s✳ ❙❡❥❛♠G✉♠ ❣r✉♣♦ ❡X ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡G✳ ❉❡♥♦t❛♠♦s ♣♦rhXi ♦ s✉❜❣r✉♣♦ ❞❡

G ❣❡r❛❞♦ ♣♦r X✳ ❙❡ X é ✜♥✐t♦ ❡ ❣❡r❛G✱G=hXi✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡ Gé ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳

❯♠❛ ♣❛❧❛✈r❛ s♦❜r❡X✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦ n✱ é ✉♠ ❡❧❡♠❡♥t♦ω ∈G ❞❛ ❢♦r♠❛

ω=xe1₁ xe2₂ . . . xen

n ,

♦♥❞❡ xi∈X✱ei ∈ {−1,1}❡ n≥1✳

❙❡❥❛♠ x, y ∈ G✳ ❖ ❝♦♠✉t❛❞♦r ❞❡ x ❡ y é ❞❡✜♥✐❞♦ ❝♦♠♦ [x, y] = x−1y−1xy✳ ❙❡❥❛♠ H, K

s✉❜❣r✉♣♦s ❞❡ G✱ ❞❡✜♥✐♠♦s ♦ ❝♦♠✉t❛❞♦r ❞❡H ❡ K ❝♦♠♦ s❡♥❞♦ ♦ s✉❜❣r✉♣♦

✻

❆❧é♠ ❞✐ss♦✱ s❡ H=K=G✱ ❡♥tã♦ G′ = [G, G]é ❞✐t♦ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞❡G✳

❉❡✜♥✐çã♦ ✶✳✶✸✳ ❆ sér✐❡ ❝❡♥tr❛❧ ✐♥❢❡r✐♦r ❞❡ ✉♠ ❣r✉♣♦ Gé ❞❡✜♥✐❞❛ ♣♦r✿

γ1(G) := G

γ2(G) := [γ1(G), G] ✳✳✳

γr(G) := [γr−1(G), G]

♦♥❞❡ r ≥1✳

❉✐③❡♠♦s q✉❡ G é ♥✐❧♣♦t❡♥t❡ ❞❡ ❝❧❛ss❡ r✱ s❡ r é ♦ ♠❡♥♦r ♥ú♠❡r♦ ♥❛t✉r❛❧ t❛❧ q✉❡ γr+1(G) = 1✳

❆ ❞❡♠♦♥str❛çã♦ ❞♦ ♣ró①✐♠♦ t❡♦r❡♠❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ✭❬✶✼❪✱ ❚❡♦r❡♠❛ ✺✳✸✻✮

❚❡♦r❡♠❛ ✶✳✶✹✳ ❙❡ Gé ♥✐❧♣♦t❡♥t❡ ❞❡ ❝❧❛ss❡r ❡ HEG ❡♥tã♦G/H é ♥✐❧♣♦t❡♥t❡ ❞❡ ❝❧❛ss❡ ♠❡♥♦r

❈❛♣ít✉❧♦

2

❉í❣r❛❢♦s ❡ ●r✉♣♦s

◆❡st❡ ❝❛♣ít✉❧♦✱ ✐♥✐❝✐❛❧♠❡♥t❡ r❡❧❡♠❜r❛r❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞♦s ❞í❣r❛❢♦s✳ ❯♠ ❞❡st❡s ♠♦str❛ q✉❡ ❡♠ ✉♠ ❞í❣r❛❢♦ ✐♥✜♥✐t♦ tr❛♥s✐t✐✈♦ ❡ ❝♦♥❡①♦ s❡♠♣r❡ ❡①✐st❡ ✉♠ ♣❛ss❡✐♦ ❞✐r❡❝✐♦♥❛❞♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡ ✐♥✐❝✐❛♥❞♦ ❡♠ q✉❛❧q✉❡r ✈ért✐❝❡✳ ◆❛ ❙❡çã♦✷✳✷❞❡✜♥✐r❡♠♦s ♣r♦♣r✐❡❞❛❞❡Z❡ ♣r♦✈❛r❡♠♦s q✉❡ ✉♠ ❞í❣r❛❢♦D✐♥✜♥✐t♦✱ ❝♦♥❡①♦✱ ❡ tr❛♥s✐t✐✈♦✱

t❡♠ ♣r♦♣r✐❡❞❛❞❡ Z s❡✱ ❡ s♦♠❡♥t❡ s❡✱ t♦❞♦ ❝✐❝❧♦ ❞❡ D é ❜❛❧❛♥❝❡❛❞♦ ✭✈❡❥❛ ▲❡♠❛ ✷✳✹✮✳ ◆❛ ❙❡çã♦

✷✳✸ ❡st✉❞❛r❡♠♦s ❞í❣r❛❢♦s ❞❡ ❈❛②❧❡②✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ❧❡♠❛ q✉❡ ♥♦s ❞á ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ✉♠ ❞í❣r❛❢♦ s❡r ✉♠ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡②✳ ❋✐♥❛❧✐③❛r❡♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠❛ s❡çã♦ s♦❜r❡ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❞í❣r❛❢♦s✳

✷✳✶ ❉❡✜♥✐çõ❡s

❯♠ ❞í❣r❛❢♦ D é ✉♠ ♣❛r ♦r❞❡♥❛❞♦ (V D, ED)✱ ♦♥❞❡ V D é ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❡

ED ⊆ V D×V D✱ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ♦r❞❡♥❛❞♦s ❞❡ V D✳ ❖s ❡❧❡♠❡♥t♦s ❞❡ V D sã♦ ❝❤❛♠❛❞♦s

❞❡ ✈ért✐❝❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ ED❞❡ ❛r❡st❛s ❞♦ ❞í❣r❛❢♦D✳ ◆♦t❡ q✉❡ ✉♠ ❞í❣r❛❢♦ ♣♦❞❡ ❝♦♥t❡r ❧❛✲

ç♦s(v, v)✱ ❜❡♠ ❝♦♠♦ ♣❛r❡s ❞❡ ❛r❡st❛s ♦♣♦st❛s(u, v)❡(v, u)✳ ❊♥❢❛t✐③❛♠♦s q✉❡ ❝♦♠ ❡st❛ ❞❡✜♥✐çã♦ ♦s ❞í❣r❛❢♦s ❝♦♥s✐❞❡r❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦ sã♦ s❡♠♣r❡ s✐♠♣❧❡s✱ ♥♦ s❡♥t✐❞♦ q✉❡ ❡♥tr❡ ❞♦✐s ✈ért✐❝❡s ❡①✐st❡ ♥♦ ♠á①✐♠♦ ✉♠❛ ❛r❡st❛ ❡♠ ❝❛❞❛ ❞✐r❡çã♦✳ ❉✐③❡♠♦s q✉❡ D′ = (V D′, ED′) é ✉♠ s✉❜❞í❣r❛❢♦ ❞❡ D s❡V D′ ⊆V D ❡ED′ ⊆ED✳

✷✳✶ ❉❡✜♥✐çõ❡s ✽

q✉❡ Dé ✐♥✜♥✐t♦✳

❊①❡♠♣❧♦ ✷✳✶✳ ❆ ❋✐❣✉r❛ ✷✳✶ ❛♣r❡s❡♥t❛ ✉♠ ❞í❣r❛❢♦ ✜♥✐t♦ ❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s

V D={1,2,3,4,5,6,7}

❡ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s

ED={(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,1)}.

❋✐❣✉r❛ ✷✳✶✿ ❡①❡♠♣❧♦ ❞❡ ❞í❣r❛❢♦

❊①❡♠♣❧♦ ✷✳✷✳ ❖ ❞í❣r❛❢♦ D ❞❛ ❋✐❣✉r❛ ✷✳✷é ✐♥✜♥✐t♦✱ t❡♠ ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ♦ ❝♦♥❥✉♥t♦

❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s Z❡ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛sED={(i, i+1)|i∈Z}✳ ❖ ❞í❣r❛❢♦Dé ❝♦♥❤❡❝✐❞♦

❝♦♠♦ ❞í❣r❛❢♦ Z ♦✉ ❞í❣r❛❢♦ ✐♥t❡✐r♦✳

❋✐❣✉r❛ ✷✳✷✿ ❞í❣r❛❢♦ ❩

❉❛❞♦ ✉♠ ❞í❣r❛❢♦D✱ ♦ ❣r❛❢♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡Dé ❛ ❡str✉t✉r❛ ♦❜t✐❞❛ ❞❡D❞❡s❝♦♥s✐❞❡r❛♥❞♦

✷✳✶ ❉❡✜♥✐çõ❡s ✾

❯♠ ♣❛ss❡✐♦W = (v0, ǫ1, v1, . . . , ǫn, vn)❞❡v0❛vn✱ ❞❡ ❝♦♠♣r✐♠❡♥t♦|W|=n✱ é ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ n+ 1✈ért✐❝❡s ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❞✐st✐♥t♦s✮ v0, v1, . . . , vn ❡ ❞❡n✐♥❞✐❝❛❞♦r❡sǫ1, ǫ2, . . . , ǫn∈

{1,−1} t❛❧ q✉❡ ♣❛r❛ t♦❞♦j∈ {1,2, . . . , n}t❡♠♦s✿

ǫj = 1⇒(vj−1, vj)∈EW

ǫj =−1⇒(vj, vj−1)∈EW.

■♥t✉✐t✐✈❛♠❡♥t❡✱ ✉♠ ♣❛ss❡✐♦ é ✉♠ ♣❡r❝✉rs♦ ♥♦ ❞í❣r❛❢♦ ❞❡ ✈ért✐❝❡ ❡♠ ✈ért✐❝❡ ❛♦ ❧♦♥❣♦ ❞❛s ❛r❡st❛s✱ ♦♥❞❡ ♦s ✐♥❞✐❝❛❞♦r❡s 1❡ −1 ✐♥❢♦r♠❛♠ q✉❛♥❞♦ ♦ ♣❡r❝✉rs♦ r❡s♣❡✐t❛✱ ♦✉ ♥ã♦✱ ♦ s❡♥t✐❞♦ ❞❛ ❛r❡st❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❉✐③❡♠♦s q✉❡ v0 é ♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧ ❞❡ W ❡ vn é ✈ért✐❝❡ ✜♥❛❧ ❞❡W✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ W é ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ s❡ v0=vn✳

❯♠ ♣❛ss❡✐♦ é ❞✐r❡❝✐♦♥❛❞♦ s❡ t♦❞♦s ♦s ✐♥❞✐❝❛❞♦r❡s ❞❡W sã♦ ✐❣✉❛✐s à ✶ ✭♦✉ ✲✶✮✱ ❡ ❡❧❡ é ❛❧t❡r♥❛❞♦

s❡ ♦s ✈❛❧♦r❡s ❞♦s ✐♥❞✐❝❛❞♦r❡s ❛❧t❡r♥❛♠✳ ❉❛❞♦ ✉♠ ♣❛ss❡✐♦ W = (v0, ǫ1, v1, . . . , ǫn, vn)✱ ♦ ♣❛ss❡✐♦ ✐♥✈❡rs♦ ❞❡W éW−1 = (vn,−ǫn, vn−1, . . . ,−ǫ1, v0)✳ P❛r❛0≤i≤j≤n✱ ❛ s✉❜s❡q✉ê♥❝✐❛

iWj = (vi, ǫi+1, vi+1, . . . , ǫj, vj)

é ❞✐t❛ ✉♠ s✉❜♣❛ss❡✐♦ ❞❡ W✳ ❆❧é♠ ❞✐ss♦✱ s❡

W′ = (u0, δ1, u1, . . . , δm, um)

é ✉♠ ♣❛ss❡✐♦ t❛❧ q✉❡ u0=vn✱ ❡♥tã♦ ❛ ❝♦♥❝❛t❡♥❛çã♦ ❞❡ W ❡ W′ é ♦ ♣❛ss❡✐♦

W.W′ = (v0, ǫ1, v1, . . . , ǫn, u0, δ1, u1, . . . , δm, um)

❞❡ ❝♦♠♣r✐♠❡♥t♦ n+m✳ ❚❡♠♦s t❛♠❜é♠ q✉❡ ♦ ♣❡s♦ ❞❡ W✱ ❞❡♥♦t❛❞♦ ♣♦rF(W)✱ é

F(W) =ǫ1+ǫ2+· · ·+ǫn,

✷✳✶ ❉❡✜♥✐çõ❡s ✶✵

❯♠ ❝❛♠✐♥❤♦ é ✉♠ ♣❛ss❡✐♦ W ❝✉❥♦s ✈ért✐❝❡s sã♦ ❞♦✐s ❛ ❞♦✐s ❞✐st✐♥t♦s✳ ❉✐③❡♠♦s q✉❡ ♦ ❞í❣r❛❢♦

D é ❝♦♥❡①♦ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ✈ért✐❝❡s u, v ∈ V D ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❧✐❣❛♥❞♦ u ❛ v✳ ❈❛s♦

❝♦♥trár✐♦✱ ❞✐③❡♠♦s q✉❡ Dé ❞❡s❝♦♥❡①♦✳ ❯♠ ❝✐❝❧♦C é ✉♠ ❝❛♠✐♥❤♦ ❝✉❥♦s ❡①tr❡♠♦s ❝♦✐♥❝✐❞❡♠✳ ❙❡

D é ❝♦♥❡①♦ ❡ ♥ã♦ ❝♦♥té♠ ❝✐❝❧♦s✱ ❡♥tã♦ ❞✐③❡♠♦s q✉❡Dé ✉♠❛ ár✈♦r❡✳

❙❡❥❛ D ✉♠ ❞í❣r❛❢♦✳ ❉❛❞♦ v ∈ V D ❡ j ∈ N✳ ❉❡✜♥✐♠♦s D_j+(v) ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ✜♥❛✐s ❞❡ t♦❞♦s ♣❛ss❡✐♦s ❞✐r❡❝✐♦♥❛❞♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ j ❝♦♠ ✈ért✐❝❡ ✐♥✐❝✐❛❧ v ❡ D_j−(v) ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈ért✐❝❡s ✐♥✐❝✐❛✐s ❞❡ t♦❞♦s ♦s ♣❛ss❡✐♦s ❞✐r❡❝✐♦♥❛❞♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ j ❝♦♠

✈ért✐❝❡ ✜♥❛❧ v✳ ❙❡S ⊂V D❡♥tã♦

D_j+(S) = [

v∈S

D+_j (v) ❡ D_j−(S) = [

v∈S

D−_j (v).

❖ ❣r❛✉ ❞❡ s❛í❞❛ d+_{(v) ❞❡ vé ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ D}+

1(v)✳ ❊ ♦ ❣r❛✉ ❞❡ ❡♥tr❛❞❛ d−(v) ❞❡

v é ❛ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡D−₁(v)✳ ❙❡ ♦s ❣r❛✉s ❞❡ s❛í❞❛ ✭r❡s♣❡❝t✐✈❛♠❡♥t❡ ❞❡ ❡♥tr❛❞❛✮ sã♦ ♦s ♠❡s♠♦s

♣❛r❛ t♦❞♦ ✈ért✐❝❡ ❞❡D✱ ❡s❝r❡✈❡r❡♠♦s ❛♣❡♥❛sd+_D ✭r❡s♣❡❝t✐✈❛♠❡♥t❡d−_D✮✳ ❖ s✉❜í♥❞✐❝❡ Dé ♦♠✐t✐❞♦

s❡D❡stá ❝❧❛r♦ ♥♦ ❝♦♥t❡①t♦✳ ❯♠ ❞í❣r❛❢♦ é ❞✐t♦ r❡❣✉❧❛r s❡d+(v) =d+(v′)❡ d−(v) =d−(v′)✱ ♣❛r❛ t♦❞♦ v, v′ ∈V D✳

❯♠ ❞í❣r❛❢♦Dé ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦ s❡ ❛♠❜♦sd+(v)❡d−(v)sã♦ ✜♥✐t♦s ♣❛r❛ t♦❞♦v∈V D✳ ❈❛s♦

❝♦♥trár✐♦ ❞✐r❡♠♦s q✉❡ Dé ❧♦❝❛❧♠❡♥t❡ ✐♥✜♥✐t♦✳

❆♥t❡s ❞❡ ✜♥❛❧✐③❛r♠♦s ❡st❛ s❡çã♦✱ ❞❡✜♥✐r❡♠♦s ❛✉t♦♠♦r✜s♠♦ ❞❡ ❞í❣r❛❢♦s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❞í❣r❛❢♦ tr❛♥s✐t✐✈♦✳

❙❡❥❛♠ D ❡ D′ _{❞í❣r❛❢♦s✳ ❯♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❞í❣r❛❢♦s φ : D → D}′ _{é ✉♠❛ ❛♣❧✐❝❛çã♦}

φ:V D→V D′ q✉❡ ♣r❡s❡r✈❛ ❛r❡st❛✱ ✐st♦ é✱

(u, v)∈ED⇒(φ(u), φ(v))∈ED′.

❙❡ φ é s♦❜r❡❥❡t♦r❛ ❡♥tã♦ φé ✉♠ ❡♣✐♠♦r✜s♠♦✳ ❙❡ φé ❜✐❥❡t♦r ❡ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ♣r❡s❡r✈❛ ❛r❡st❛✱

❡♥tã♦ φ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡ ❞✐③❡♠♦s q✉❡ D ❡ D′ sã♦ ✐s♦♠♦r❢♦s✳ ❊✱ s❡ φ é ✉♠ ✐s♦♠♦r✜s♠♦ ❡

D=D′✱ ❡♥tã♦φé ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ ❞í❣r❛❢♦s✳

✷✳✷ Pr♦♣r✐❡❞❛❞❡ ❩ ✶✶

s♦❜r❡ V D✱ ❞✐③❡♠♦s q✉❡ ♦ ❞í❣r❛❢♦Dé tr❛♥s✐t✐✈♦✳ ❖❜s❡r✈❡ q✉❡ ❛ ❛çã♦ ❞❡Gs♦❜r❡ V D✐♥❞✉③ ✉♠❛

❛çã♦ ♥❛t✉r❛❧ s♦❜r❡ ED ❞❛❞❛ ♣♦r✿

g_{(u, v) := (}g_u, g_v),

♣❛r❛ t♦❞♦ (u, v)∈ED❡ g∈G✳

▲❡♠❛ ✷✳✸✳ ❙❡❥❛ D ✉♠ ❞í❣r❛❢♦ ✐♥✜♥✐t♦✱ ❝♦♥❡①♦ ❡ tr❛♥s✐t✐✈♦✳ ❊♥tã♦ ❡①✐st❡♠ ❡♠ D ❝❛♠✐♥❤♦s

❞✐r❡❝✐♦♥❛❞♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡ ✳

❉❡♠♦♥str❛çã♦✳ ❙❡ Dé ❧♦❝❛❧♠❡♥t❡ ✜♥✐t♦✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❝❛♠✐♥❤♦s ❞✐r❡❝✐♦♥❛❞♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦

❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡ é ❣❛r❛♥t✐❞❛ ♣❡❧♦ r❡s✉❧t❛❞♦ ❞❡ ❚r♦✜♠♦✈ ❡♠ ❬✶✾❪✳

❙❡ Dé ❧♦❝❛❧♠❡♥t❡ ✐♥✜♥✐t♦✱ ❡♥tã♦ ♣❡❧♦ ♠❡♥♦sd+ ♦✉d− é ✐♥✜♥✐t♦✳ ❙✉♣♦♥❤❛ q✉❡d+ é ✐♥✜♥✐t♦

❡ s❡❥❛ v1 ∈ V D✳ ❉❛ s✉♣♦s✐çã♦ ❡ ❞❡ D ❝♦♥❡①♦✱ ❡①✐st❡ v2 ∈ V D t❛❧ q✉❡ (v1, v2) ∈ED✳ ❊s❝♦❧❤❛ ❛❣♦r❛ ✉♠ v3 ∈ V D t❛❧ q✉❡ (v2, v3)∈ ED✱ ♦♥❞❡ v1 6=v2 6=v3 ✭v3 ❡①✐st❡ ♣♦✐s d+ é ✐♥✜♥✐t♦✱ ✐st♦ é✱ ♣♦❞❡♠♦s s❡♠♣r❡ ❡s❝♦❧❤❡r ✈ért✐❝❡s ❞✐st✐♥t♦s✮✳ ❘❡♣❡t✐♥❞♦ ❡st❡ ♣r♦❝❡ss♦ q✉❛♥t❛s ✈❡③❡s ❢♦r❡♠ ♥❡❝❡ssár✐❛s✱ ♦❜t❡♠♦s ♦ ❝❛♠✐♥❤♦ ❞❡s❡❥❛❞♦✳ ❉❡ ❢❛t♦✱ ✉♠❛ ✈❡③ q✉❡ ❡♠ ❝❛❞❛ ♣❛ss♦ t❡♠♦s ✐♥✜♥✐t❛s ♦♣çõ❡s ❞❡ ❡s❝♦❧❤❛s ❞❡ ✈ért✐❝❡s ❡ ❛♣ós t❛❧ ♣❛ss♦ t❡♠♦s s♦♠❡♥t❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈ért✐❝❡s ✈✐s✐t❛❞♦s✳ ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ♣r♦✈❛✲s❡ ♦ ❝❛s♦ d− =∞✳

✷✳✷ Pr♦♣r✐❡❞❛❞❡ ❩

❙❡❥❛ D✉♠ ❞í❣r❛❢♦✳ ❉✐③❡♠♦s q✉❡D t❡♠ ♣r♦♣r✐❡❞❛❞❡ ❩ s❡ ❡①✐st❡ ✉♠ ❡♣✐♠♦r✜s♠♦φ:D→Z✱

♦♥❞❡ Z é ♦ ❞í❣r❛❢♦ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❊①❡♠♣❧♦✷✳✷✳

P❛r❛ ✉♠ ❞í❣r❛❢♦ ❝♦♥❡①♦✱ ✐♥✜♥✐t♦ ❡ tr❛♥s✐t✐✈♦ D t❡r ❛ ♣r♦♣r✐❡❞❛❞❡ ❩ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❛✜r♠❛r

q✉❡ t♦❞♦s ♦s ❝✐❝❧♦s ❞❡ Dsã♦ ❜❛❧❛♥❝❡❛❞♦s✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ♥♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✳

▲❡♠❛ ✷✳✹✳ ❙❡❥❛ D ✉♠ ❞í❣r❛❢♦ ✐♥✜♥✐t♦✱ ❝♦♥❡①♦ ❡ tr❛♥s✐t✐✈♦✳ ❊♥tã♦ D t❡♠ ♣r♦♣r✐❡❞❛❞❡ ❩ s❡✱ ❡

s♦♠❡♥t❡ s❡✱ t♦❞♦ ❝✐❝❧♦ ❞❡ D é ❜❛❧❛♥❝❡❛❞♦✳

❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ D t❡♠ ♣r♦♣r✐❡❞❛❞❡ ❩✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ s♦❜r❡❥❡t♦r

φ : D → Z✳ ❙❡❥❛ C = (v0, ǫ1, v1, . . . , ǫn, vn) ✉♠ ♣❛ss❡✐♦ s✐♠♣❧❡s ❢❡❝❤❛❞♦ ❞❡ D✳ ❈♦♠♦ φ é ✉♠

❤♦♠♦♠♦r✜s♠♦✱ ❡♥tã♦

φ(vn) =φ(v0) +

n X

i=1

✷✳✸ ❉í❣r❛❢♦s ❞❡ ❈❛②❧❡② ✶✷

❈♦♠♦C é ✉♠ ♣❛ss❡✐♦ ❢❡❝❤❛❞♦ s❡❣✉❡ q✉❡v0 =vn✳ ▲♦❣♦φ(vn) =φ(v0) ❡✱ ♣♦rt❛♥t♦F(C) = 0✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ✜①❡ u ∈ V D✳ ❉❡✜♥❛ ✉♠❛ ❛♣❧✐❝❛çã♦ φ : V D → V Z t❛❧ q✉❡ φ(u) = 0 ❡

φ(v) :=F(W)✱ ♦♥❞❡W é ✉♠ ♣❛ss❡✐♦ ❞❡u❛tév✳ ❖❜s❡r✈❡ q✉❡✱ s❡ ❡①✐st❡♠ ❞♦✐s ❝❛♠✐♥❤♦s ❞✐st✐♥t♦s

❧✐❣❛♥❞♦ u à v✱ ❞✐❣❛♠♦s W1 ❡ W2✱ ❡♥tã♦ ♦ ♣❛ss❡✐♦ W = W1.(W2)−1 é ❢♦r♠❛❞♦ ♣❡❧❛ ✉♥✐ã♦ ❞❡ ♣❛ss❡✐♦s s✐♠♣❧❡s ❢❡❝❤❛❞♦sC1, C2, . . . , Cn✳ ▲♦❣♦✱F(W) = 0♣♦✐sF(Ci) = 0♣♦r ❤✐♣ót❡s❡✳ P♦rt❛♥t♦

F(W1) = −F(W₂−1) =F(W2)✱ ❡ s❡❣✉❡ q✉❡ φ❡stá ❜❡♠ ❞❡✜♥✐❞❛✳ ❆❣♦r❛✱ s❡❥❛ (x, y)∈ ED✳ P♦r

❞❡✜♥✐çã♦✱φ(y) =φ(x) + 1✳ ❈♦♠♦φ(x) =i♣❛r❛ ❛❧❣✉♠i∈Z✱ s❡❣✉❡ q✉❡φ(y) =i+ 1✱ ❡ ♣♦rt❛♥t♦✱

(φ(x), φ(y)) = (i, i+ 1) ∈ EZ✳ ▲♦❣♦✱ φ é ✉♠ ❤♦♠♦♠♦r✜s♠♦✳ ❋✐♥❛❧♠❡♥t❡✱ ♦ ❢❛t♦ ❞❡ φ s❡r

s♦❜r❡❥❡t♦r❛ s❡❣✉❡ ❞♦ ▲❡♠❛ ✷✳✸✳

❊①❡♠♣❧♦ ✷✳✺✳ ❯♠❛ ár✈♦r❡ ✐♥✜♥✐t❛ r❡❣✉❧❛r ✭❝♦♠ ❣r❛✉ ❞❡ ❡♥tr❛❞❛ ❡ ❣r❛✉ ❞❡ s❛í❞❛ ❝♦♥st❛♥t❡✮ ❝❧❛r❛♠❡♥t❡ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ Z✱ ♣♦✐s ♥ã♦ ❝♦♥té♠ ❝✐❝❧♦s✳

❊♠ ❬✶✻❪✱ Pr❛❡❣❡r ❞á ✉♠❛ ❝♦♥❞✐çã♦ ♣❛r❛ ✉♠ ❞í❣r❛❢♦ ❝♦♥❡①♦✱ ✐♥✜♥✐t♦✱ tr❛♥s✐t✐✈♦ ♥♦s ✈ért✐❝❡s ❡ ♥❛s ❛r❡st❛s t❡r ♣r♦♣r✐❡❞❛❞❡ Z✳ ❈♦♠♦ ♣♦❞❡♠♦s ✈❡r ❛ s❡❣✉✐r✳

❚❡♦r❡♠❛✳ ❙❡❥❛ D ✉♠ ❞í❣r❛❢♦ ✐♥✜♥✐t♦✱ ❝♦♥❡①♦✱ tr❛♥s✐t✐✈♦ ♥♦s ✈ért✐❝❡s ❡ ♥❛s ❛r❡st❛s ❡ ❝♦♠ ❣r❛✉

❞❡ ❡♥tr❛❞❛ ❡ ❣r❛✉ ❞❡ s❛í❞❛ ❛♠❜♦s ✜♥✐t♦s ♠❛s ❞✐st✐♥t♦s✳ ❊♥tã♦ D t❡♠ ♣r♦♣r✐❡❞❛❞❡Z✳

✷✳✸ ❉í❣r❛❢♦s ❞❡ ❈❛②❧❡②

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❛ s❡çã♦ s❡rá ❞✐s❝✉t✐r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❜ás✐❝♦s s♦❜r❡ ❞í❣r❛❢♦s ❞❡ ❈❛②❧❡②✳ ❖ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② Cay(G, S) ❞❡ ✉♠ ❣r✉♣♦ G ❝♦♠ r❡s♣❡✐t♦ à ✉♠ s✉❜❝♦♥❥✉♥t♦ S ❞❡

G\ {1}✱ ♦♥❞❡1❞❡♥♦t❛ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡G✱ t❡♠ ❝♦♠♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ♦ ❣r✉♣♦G❡ ♦ ❝♦♥❥✉♥t♦

❞❡ ❛r❡st❛s ECay(G, S) ={(g, gs)|s∈S}✳

◆♦t❡ q✉❡ G ❛❣❡ r❡❣✉❧❛r♠❡♥t❡ s♦❜r❡ D = Cay(G, S) ♣♦r ♠✉❧t✐♣❧✐❝❛çã♦ à ❡sq✉❡r❞❛✳ ❉❛❞♦s

g, x ∈G✱ x_{g := xg✱ ♣❛r❛ t♦❞♦ g ∈G✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❡♠♦s ✉♠❛ ❛çã♦ ♥❛t✉r❛❧ ❞❡ G s♦❜r❡}

❛s ❛r❡st❛s ❞❡ D❞❛❞❛ ♣♦r✿

x_{(g, gs) = (}x_g,(x_{g)s) = (xg, xgs),}

♦♥❞❡ x∈G✳ ❈♦♠♦(xg, x(gs))∈ED✱ s❡❣✉❡ q✉❡G♣r❡s❡r✈❛ ❛r❡st❛ ❡✱ ♣♦rt❛♥t♦Gé ✉♠ s✉❜❣r✉♣♦

✷✳✸ ❉í❣r❛❢♦s ❞❡ ❈❛②❧❡② ✶✸

▲❡♠❛ ✷✳✻✳ ❙❡❥❛ D = Cay(G, S) ✉♠ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② ❝♦♠ r❡s♣❡✐t♦ ❛♦ s✉❜❝♦♥❥✉♥t♦ S ❞❡ G✳

❊♥tã♦ D é ❝♦♥❡①♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ S é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ❞❡G✳

❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ♦❜s❡r✈❡ q✉❡ q✉❛❧q✉❡r ❝❛♠✐♥❤♦ ❡♠ D❝✉❥♦ ✈ért✐❝❡ ✐♥✐❝✐❛❧ é ❛ ✐❞❡♥✲

t✐❞❛❞❡✱ t❡♠ ❝♦♠♦ ✈ért✐❝❡ ✜♥❛❧ ✉♠❛ ♣❛❧❛✈r❛ ❡♠ S✳ ❙✉♣♦♥❤❛ q✉❡ S é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s

❞❡ G✳ ❊♥tã♦ ♣❛r❛ q✉❛✐sq✉❡r ❞♦✐s ✈ért✐❝❡su ❡ v❞❡ D✱ t❡♠♦s q✉❡u ❡v sã♦ ♣❛❧❛✈r❛s ❡♠S✳ ▲♦❣♦✱

❡①✐st❡♠ ❡♠D✉♠ ❝❛♠✐♥❤♦W ❞❛ ✐❞❡♥t✐❞❛❞❡ ❛téu✱ ❡ ✉♠ ❝❛♠✐♥❤♦W′ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❛tév✳ ❆ss✐♠✱

❛ ❝♦♥❝❛t❡♥❛çã♦ W−1.W′ é ✉♠ ♣❛ss❡✐♦ ❞❡ u ❛ v✳ ❈♦♠♦✱ u ❡ v sã♦ ❛r❜✐trár✐♦s✱ s❡❣✉❡ q✉❡ D é

❝♦♥❡①♦✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡❥❛ v ∈ V D✳ ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡ v é ✉♠❛ ♣❛❧❛✈r❛ ❡♠ S✳ ❈♦♠♦ D

é ❝♦♥❡①♦✱ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ❝♦♠ ✈ért✐❝❡ ✐♥✐❝✐❛❧ ✐❣✉❛❧ ❛ ✐❞❡♥t✐❞❛❞❡ ❡ ✈ért✐❝❡ ✜♥❛❧ v✱ ♠❛s ♣❡❧❛

♦❜s❡r✈❛çã♦ ❛❝✐♠❛ t❡♠♦s q✉❡ vé ✉♠❛ ♣❛❧❛✈r❛ ❡♠S✳ ❈♦♠♦vé ❛r❜✐trár✐♦✱ ❡♥tã♦S é ✉♠ ❝♦♥❥✉♥t♦

❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ G✳

❊♥❢❛t✐③❛♠♦s q✉❡✱ s❛❧✈♦ ♠❡♥çã♦ ❝♦♥trár✐❛✱ ❡st❛r❡♠♦s ❝♦♥s✐❞❡r❛♥❞♦ q✉❡Cay(G, S)é ✉♠ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② ❞❡ G❝♦♠ r❡s♣❡✐t♦ ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡sS ❞❡G✳

❊①❡♠♣❧♦ ✷✳✼✳ ◆❛ ❋✐❣✉r❛ ✷✳✸ ❛♣r❡s❡♥t❛♠♦s ♦ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② Cay(S3, S)✱ ♦♥❞❡

S3=hr, t|r3 = 1, t2= 1, rtr=ti ❡ S={r, t}✳

❋✐❣✉r❛ ✷✳✸✿ Cay(S3, S)✱ ❛r❡st❛ ❛③✉❧✿ ❣❡r❛❞❛ ♣♦rt❀ ❛r❡st❛ ✈❡r♠❡❧❤❛✿ ❣❡r❛❞❛ ♣♦r r✳

❊①❡♠♣❧♦ ✷✳✽✳ ◆❛ ✜❣✉r❛ ✷✳✹ ❛♣r❡s❡♥t❛♠♦s ♦ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② Cay(D4, S)✱ ♦♥❞❡

✷✳✸ ❉í❣r❛❢♦s ❞❡ ❈❛②❧❡② ✶✹

❋✐❣✉r❛ ✷✳✹✿ Cay(D4, S)✱ ❛r❡st❛ ❛③✉❧✿ ❣❡r❛❞❛ ♣♦rt❀ ❛r❡st❛ ✈❡r♠❡❧❤❛✿ ❣❡r❛❞❛ ♣♦r r✳

▲❡♠❛ ✷✳✾✳ ❙❡❥❛ D ✉♠ ❞í❣r❛❢♦ ❝♦♥❡①♦✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠ s✉❜❣r✉♣♦ G ❞❡ Aut(D) ❛❣✐♥❞♦ r❡❣✉❧❛r♠❡♥t❡ s♦❜r❡ D✳ ❊♥tã♦ D é ✐s♦♠♦r❢♦ ❛♦ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② Cay(G, S) ❞❡ G ♣❛r❛ ❛❧❣✉♠

❝♦♥❥✉♥t♦ S ❞❡ ❣❡r❛❞♦r❡s ❞❡ G✳

❉❡♠♦♥str❛çã♦✳ ❋✐①❡ v ∈ V D✳ ❈♦♠♦ G ❛❣❡ r❡❣✉❧❛r♠❡♥t❡✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ u ∈ V D ❡①✐st❡ ✉♠

ú♥✐❝♦ g∈G t❛❧ q✉❡u= g_{v✳ ❚♦♠❡}

S = {g∈G|(v, gv)∈ED}.

Pr✐♠❡✐r❛♠❡♥t❡✱ ♠♦str❛r❡♠♦s q✉❡ S ❣❡r❛ G✳ ❙❡❥❛ g ∈ G ❡ ❝♦♥s✐❞❡r❡ ♦ ✈ért✐❝❡ g_{v✳ ❈♦♠♦ D é}

❝♦♥❡①♦✱ ❡①✐st❡

P = (v0, v1, . . . , vn−1, vn)

✉♠ ❝❛♠✐♥❤♦ ❞❡v ❛té g_{v✳ ❱❛♠♦s ♣r♦✈❛r ♣♦r ✐♥❞✉çã♦ s♦❜r❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❝❛♠✐♥❤♦P✱ q✉❡g}

é ✉♠❛ ♣❛❧❛✈r❛ ❡♠ S✳ ❙❡ n= 1✱ ❡♥tã♦ g_{v∈ {}x_{v |x∈S∪S}−1_}✳

❆ss✉♠❛ ❛❣♦r❛n≥2❡ s✉♣♦♥❤❛ q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈á❧✐❞♦ ♣❛r❛ t♦❞♦ ❝❛♠✐♥❤♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦

l < n✳ ❈♦♠♦ 0Pn−1 é ✉♠ ❝❛♠✐♥❤♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦n−1✱ ♣♦r ✐♥❞✉çã♦✱vn−1= ωv✱ ♣❛r❛ ❛❧❣✉♠❛

♣❛❧❛✈r❛ ω ❡♠ S✳ ❈♦♠ ✐st♦ t❡♠♦s ❞♦✐s ❝❛s♦s✳ Pr✐♠❡✐r♦✱ s❡ (vn−1, gv) = (ωv, gv) é ✉♠❛ ❛r❡st❛ ❡♠ D ❡♥tã♦ (v, ω−1_g

v) t❛♠❜é♠ é ✉♠❛ ❛r❡st❛ ❡♠ D✱ ❧♦❣♦ ω−1g ∈ S✳ ❆ss✐♠ ❡①✐st❡ s ∈ S t❛❧

q✉❡ ω−1g =s✱ ❡ ✐st♦ ✐♠♣❧✐❝❛ q✉❡ g =ωs✳ ❈♦♠♦s, ω ∈S✱ ♦❜t❡♠♦s q✉❡ g é ✉♠❛ ♣❛❧❛✈r❛ ❡♠S✳

Pr♦✈❛✲s❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ s❡(g_{v, vn−}

✷✳✸ ❉í❣r❛❢♦s ❞❡ ❈❛②❧❡② ✶✺

❙❡❥❛ D′=Cay(G, S) ♦ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② ❞❡ G❝♦♠ r❡s♣❡✐t♦ àS✳ ❉❡✜♥❛ ❛ ❛♣❧✐❝❛çã♦

φ:V D → V D′

g_{v → g}

▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡ φ é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❞í❣r❛❢♦✳ ❈♦♠♦ G ❛❣❡ r❡❣✉❧❛r♠❡♥t❡✱ s❡❣✉❡

q✉❡ φ é ✉♠❛ ❜✐❥❡çã♦✳ P❡❧❛ tr❛♥s✐t✐✈✐❞❛❞❡ ❞❡ G s♦❜r❡ V D✱ t❡♠♦s q✉❡ ✉♠❛ ❛r❡st❛ ❡♠ D é

❞❛ ❢♦r♠❛ (g_v, gs_{v)✱ ♣❛r❛ ❛❧❣✉♠ g ∈ G ❡ s ∈ S✳ ❈♦♠♦✱ φ(}g_{v) = g ❡ φ(}gs_{v) = gs✱ ❡♥tã♦}

(φ(g_{v), φ(}gs_{v)) = (g, gs) é ✉♠❛ ❛r❡st❛ ❡♠ D}′_{✱ ♣♦rt❛♥t♦ φ é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❞í❣r❛❢♦✳ ❉❡}

❢♦r♠❛ s✐♠✐❧❛r✱ ♦❜t❡♠♦s q✉❡ φ−1 ♣r❡s❡r✈❛ ❛r❡st❛✳ ■st♦ ♣r♦✈❛ ♦ r❡s✉❧t❛❞♦✳

❙❡❣✉❡ ❞♦ ❧❡♠❛ ❛❝✐♠❛ q✉❡✿ ❯♠ ❞í❣r❛❢♦ ❝♦♥❡①♦ D é ✉♠ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② s❡✱ ❡ s♦♠❡♥t❡ s❡✱

❡①✐st❡ H≤Aut(D) ❛❣✐♥❞♦ r❡❣✉❧❛r♠❡♥t❡ s♦❜r❡D✳

❈♦r♦❧ár✐♦ ✷✳✶✵✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❛❣✐♥❞♦ tr❛♥s✐t✐✈❛♠❡♥t❡ s♦❜r❡ ✉♠ ❞í❣r❛❢♦ D✳ ❊♥tã♦

D é ✉♠ ❞í❣r❛❢♦ ❞❡ ❈❛②❧❡② ❞❡ G❝♦♠ r❡s♣❡✐t♦ ❛ ❛❧❣✉♠ s✉❜❝♦♥❥✉♥t♦ S ⊆G\ {1}✳

❉❡♠♦♥str❛çã♦✳ P❡❧❛ ♦❜s❡r✈❛çã♦ ✶✳✹✱ G ❛❣❡ r❡❣✉❧❛r♠❡♥t❡ s♦❜r❡ D✳ ❙❡ D é ❝♦♥❡①♦✱ ♦ r❡s✉❧t❛❞♦

s❡❣✉❡ ❞♦ ▲❡♠❛ ✷✳✾✳

❆❣♦r❛✱ s✉♣♦♥❤❛ q✉❡D é ❞❡s❝♦♥❡①♦✳ ❉❛❞♦u ∈V D✱ ❞❡♥♦t❡ ♣♦rDu ❛ ❝♦♠♣♦♥❡♥t❡ ❞❡D q✉❡

❝♦♥té♠ u✳ P❡❧♦ ▲❡♠❛ ✷✳✾✱Du=Cay(hSui, Su)✱ ♦♥❞❡

Su = {g∈G|(u, gu)∈ED }.

❆❣♦r❛✱ s❡❥❛ v∈V D t❛❧ q✉❡ Dv 6=Du✳ ❈♦♠♦ Gé tr❛♥s✐t✐✈♦✱ ❡①✐st❡ g∈G t❛❧ q✉❡ v= gu✳ ❙❡❥❛

(v, s′

v)∈ED✱ ❝♦♠s′_{∈Sv✱ ❡♥tã♦ (v,} s′

v) = (g_u, s′_g

u)✳ ❉❡ G ❛❜❡❧✐❛♥♦ t❡♠♦sgs′ _=s′_{g✱ ❧♦❣♦}

(v, s′v) = (gu, gs′u) = g(u, s′u)

❝♦♠ ✐st♦✱ (u, s′

u) ∈ ED ✐♠♣❧✐❝❛ q✉❡ s′ ∈ Su✳ P♦rt❛♥t♦✱ Dv é ✉♠❛ ❝ó♣✐❛ ❞❡ Du✳ ❈♦♠♦ v é

✷✳✹ ❈r❡s❝✐♠❡♥t♦ ✶✻

✷✳✹ ❈r❡s❝✐♠❡♥t♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❡st✉❞❛r❡♠♦s ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ❞í❣r❛❢♦s tr❛♥s✐t✐✈♦s✳ ❆ ❞✐stâ♥❝✐❛ distD(u, v)

❡♥tr❡ ❞♦✐s ✈ért✐❝❡s ❡♠ ✉♠ ❞í❣r❛❢♦ ❝♦♥❡①♦ ❡ tr❛♥s✐t✐✈♦ D é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♠❡♥♦r ❝❛♠✐♥❤♦

❧✐❣❛♥❞♦u à v✳ ❚♦♠❡ v∈V D✳

❆ ❢✉♥çã♦ ❝r❡s❝✐♠❡♥t♦ f(n)✱ ❡♠ r❡❧❛çã♦ ❛ v ❝♦♠n≥0✱ é ❞❛❞❛ ♣♦r

f(n) =|{u∈V D|distD(v, u)≤n}|.

❈♦♠♦D é tr❛♥s✐t✐✈♦ ❛ ❢✉♥çã♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ ✈ért✐❝❡ v✳

❉✐③❡♠♦s q✉❡ ♦ ❞í❣r❛❢♦Dt❡♠ ❝r❡s❝✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ s❡ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ c >1 t❛❧ q✉❡

f(n)> cn

✈❛❧❡ ♣❛r❛ t♦❞♦ n >0✳ ❊ ♦ ❞í❣r❛❢♦Dt❡♠ ❝r❡s❝✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧ s❡ ❡①✐st❡♠ ❝♦♥st❛♥t❡s ♣♦s✐t✐✈❛s

c ❡ dt❛❧ q✉❡

f(n)≤cnd

✈❛❧❡ ♣❛r❛ t♦❞♦ n >0✳

❊①❡♠♣❧♦ ✷✳✶✶✳ ❆ ❋✐❣✉r❛ ✷✳✺ é ✉♠ ♣❡❞❛ç♦ ❞❡ ✉♠❛ ár✈♦r❡ T ✐♥✜♥✐t❛ r❡❣✉❧❛r ❝♦♠ d+ _{= 1 ❡}

d−_{= 2✳ ◆♦t❡ q✉❡ T é tr❛♥s✐t✐✈❛ ✭✈❡❥❛ ❬❬✶❪✱ ❚❡♦r❡♠❛ ✺✳✷✵❪✮ ❡ t❡♠♦s q✉❡}

f(1)>2; f(2)>22; f(3)>23; . . .

❈♦♥t✐♥✉❛♥❞♦ ❞❡ss❛ ♠❛♥❡✐r❛✱ ♦❜t❡♠♦s q✉❡ f(n) >2n_{✱ ♣❛r❛ t♦❞♦ n >0✳ P♦rt❛♥t♦✱ T t❡♠ ❝r❡s❝✐✲}

♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧✳

❖❜s❡r✈❛çã♦ ✷✳✶✷✳ P❛r❛ ❝❛❞❛ ✐♥t❡✐r♦ h≥0✱ ♦ s✉❜❞í❣r❛❢♦Th _{❞❡T ✐♥❞✉③✐❞♦ ❡♠} S i∈[0,h]

T_i−(v)s❡rá

❝❤❛♠❛❞♦ ❞❡ ár✈♦r❡ ❜✐♥ár✐❛ ❞❡ ❛❧t✉r❛ h ❝♦♠ r❛✐③v:=v15✳ ◆♦t❡ q✉❡

✷✳✹ ❈r❡s❝✐♠❡♥t♦ ✶✼

❋✐❣✉r❛ ✷✳✺✿ ♣❡❞❛ç♦ ❞❛ ár✈♦r❡T✳

❆ ✉♥✐ã♦ T∗ := S

i≥0

Ti _{é ✉♠❛ ár✈♦r❡ ❜✐♥ár✐❛ ❝♦♠ r❛✐③ v ❡ ❛❧t✉r❛ ❛r❜✐tr❛r✐❛♠❡♥t❡ ❣r❛♥❞❡✳}

❊①❡♠♣❧♦ ✷✳✶✸✳ ❆ ❋✐❣✉r❛ ✷✳✻ ❛♣r❡s❡♥t❛ ✉♠ ❞í❣r❛❢♦ D ❝♦♠ ❝r❡s❝✐♠❡♥t♦ ♣♦❧✐♥♦♠✐❛❧✱ ♣♦✐s

f(n)≤5n✱ ♣❛r❛ t♦❞♦ n >0✳