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✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

▼✷❘ ❈♦✉rs❡ ✏❍②♣❡r❜♦❧✐❝ ❙♣❛❝❡s ✿ ●❡♦♠❡tr② ❛♥❞ ❉✐s❝r❡t❡ ●r♦✉♣s✑ P❛rt ■ ✿ ❚❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡ ❛♥❞ ❋✉❝❤s✐❛♥ ❣r♦✉♣s

❆♥♥❡ P❛rr❡❛✉

• r❡♥♦❜❧❡✱ ❙❡♣t❡♠❜❡r ✷✵✷✵

✶✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❘❡❢❡r❡♥❝❡s

❬❇♦♥❛❤♦♥❪ ✏▲♦✇✲❉✐♠❡♥s✐♦♥❛❧ ●❡♦♠❡tr②✿❋r♦♠ ❊✉❝❧✐❞❡❛♥ ❙✉r❢❛❝❡s t♦ ❍②♣❡r❜♦❧✐❝ ❦♥♦ts✑✱ ❆▼❙✳ ✭❡❧❡♠❡♥t❛r② tr❡❛t♠❡♥t✮✳ ❬❇❡❛r❞♦♥❪ ✏❚❤❡ ❣❡♦♠❡tr② ♦❢ ❞✐s❝r❡t❡ ❣r♦✉♣s✑✱ ❙♣r✐♥❣❡r✳ ❬❘❛t❝❧✐✛❡❪ ✏❋♦✉♥❞❛t✐♦♥s ♦❢ ❍②♣❡r❜♦❧✐❝ ♠❛♥✐❢♦❧❞s✑✱ ❙♣r✐♥❣❡r✳ ❬❚❤✉rst♦♥❪ ✏❚❤r❡❡ ❞✐♠❡♥s✐♦♥❛❧ ❣❡♦♠❡tr② ❛♥❞ t♦♣♦❧♦❣②✑✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬▲♦✉st❛✉❪ ✧❍②♣❡r❜♦❧✐❝ ❣❡♦♠❡tr②✧✱ ❜♦♦❦ ✐♥ ♣r❡♣❛r❛t✐♦♥ ✭✷✵✷✵✮ ❤tt♣s✿✴✴✇✇✇✳❜r✐❝❡✳❧♦✉st❛✉✳❡✉✴r❡ss♦✉r❝❡s✴❍②♣❡r❜♦❧✐❝●❡♦♠❡tr②✳♣❞❢ ❬▼❛rt❡❧❧✐❪ ❆♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ●❡♦♠❡tr✐❝ ❚♦♣♦❧♦❣②✱ ❈r❡❛t❡❙♣❛❝❡ ■♥❞❡♣❡♥❞❡♥t P✉❜❧✐s❤✐♥❣ P❧❛t❢♦r♠ ✭✷✵✶✻✮✿ ❤tt♣✿✴✴♣❡♦♣❧❡✳❞♠✳✉♥✐♣✐✳✐t✴♠❛rt❡❧❧✐✴❣❡♦♠❡tr✐❝❴t♦♣♦❧♦❣②✳❤t♠❧ ❬❈❋❑P❪ ❈❛♥♥♦♥✲❋❧♦②❞✲❑❡♥②♦♥✲P❛rr②✱ ✏❍②♣❡r❜♦❧✐❝ ●❡♦♠❡tr②✑✱ ▼❙❘■ P✉❜✳ ❱♦❧✳ ✸✶✱ ✭✶✾✾✼✮✳ ❤tt♣✿✴✴❧✐❜r❛r②✳♠sr✐✳♦r❣✴❜♦♦❦s✴❇♦♦❦✸✶✴❢✐❧❡s✴❝❛♥♥♦♥✳♣❞❢ ❙❤♦rt r❡❢❡r❡♥❝❡ ✭✺✵ ♣❛❣❡s✮✳

✷✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈❤❛♣t❡r ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✶✳✹ ●❡♦❞❡s✐❝s ✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✸✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❉❡✜♥✐t✐♦♥

❚❤❡ ❯♣♣❡r ❍❛❧❢✲P❧❛♥❡ ✐s t❤❡ s❡t H := {(x, y) ∈ R✷, y > ✵} = {z ∈ C, Im(z) > ✵}

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✹✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❍②♣❡r❜♦❧✐❝ ❧❡♥❣t❤ ♦❢ ❛ ❝✉r✈❡✱ ■

❉❡❢s ◮ ▲❡t z ❜❡ ❛ ♣♦✐♥t ✐♥ H✳ ❚❤❡ ❤②♣❡r❜♦❧✐❝ ♥♦r♠ ❛t z ♦❢ ❛ ✈❡❝t♦r v ∈ R✷ ≃ C ✐s ❞❡✜♥❡❞ ❜② r❡s❝❛❧✐♥❣ t❤❡ ✉s✉❛❧ ❡✉❝❧✐❞❡❛♥ ♥♦r♠ v❡✉❝ = |v| ♦❢ v✿ v❤②♣

z

:= ✶ Im(z)v❡✉❝ ◮ ▲❡t c : [a, b] → H ❜❡ ❛ C ✶✲❝✉r✈❡✳ ❚❤❡ ❤②♣❡r❜♦❧✐❝ ❧❡♥❣t❤ ♦❢ c ✐s ❞❡✜♥❡❞ ❜② ✐♥t❡❣r❛t✐♥❣ t❤❡ ❤②♣❡r❜♦❧✐❝ ♥♦r♠ ♦❢ t❤❡ s♣❡❡❞ ✈❡❝t♦r✿ ℓ❤②♣(c) := b

a

c′(t)❤②♣

c(t) ❞t =

b

a

✶ y(t)

• x′(t)✷ + y′(t)✷ ❞t

✇❤❡r❡ c(t) = x(t) + iy(t)✳ ◮ ❚❤❡ ❤②♣❡r❜♦❧✐❝ ❧❡♥❣t❤ ♦❢ ❛ ♣✐❡❝❡✇✐s❡ C ✶✲❝✉r✈❡ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❤②♣❡r❜♦❧✐❝ ❧❡♥❣t❤s ♦❢ ✐ts C ✶✲♣✐❡❝❡s

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✺✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❍②♣❡r❜♦❧✐❝ ❧❡♥❣t❤✱ ■■

Pr♦♣ ℓ❤②♣(c) ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ✭✐♥❥❡❝t✐✈❡✮ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢ c✳ Pr♦♦❢ ✿ ❛♣♣❧② ❝❤❛✐♥ r✉❧❡ ✭❡①♦✮✳ ❊①❛♠♣❧❡ ❚❤❡ ❤②♣ ❧❡♥❣t❤ ♦❢ t❤❡ s❡❣♠❡♥t ❢r♦♠ iy✶ t♦ iy✷ ✐s

• ln( y✷

y✶ )

• ✳

❊①♦ ✶✳✶ ❈♦♠♣✉t❡ t❤❡ ❤②♣❡r❜♦❧✐❝ ❧❡♥❣t❤ ♦❢ t❤❡ ❤♦r✐③♦♥t❛❧ s❡❣♠❡♥t ❢r♦♠ x✶ + iy t♦ x✷ + iy✳ ✳✳✳ ❙♦❧✉t✐♦♥ ✿ ✐t ✐s ✶

✷ ✶

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✻✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥✱ ■

❉❡❢ ❚❤❡ ❤②♣❡r❜♦❧✐❝ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts z✶, z✷ ∈ H ✐s d❤②♣(z✶, z✷) := inf{ℓ❤②♣(c)} ✇❤❡r❡ t❤❡ ✐♥❢ ✐s t❛❦❡♥ ♦✈❡r ❛❧❧ ❝♦♥t✐♥✉♦✉s ♣✇ C ✶ ♣❛t❤s c ❢r♦♠ z✶ t♦ z✷ ✐♥ H✳ ❲❤❡♥ t❤❡r❡ ✐s ♥♦ ❛♠❜✐❣✉✐t② ✇❡ ✇✐❧❧ ❞❡♥♦t❡ ❢r♦♠ ♥♦✇ ♦♥ d = d❤②♣ ❛♥❞ ℓ = ℓ❤②♣ t♦ s✐♠♣❧✐❢② ♥♦t❛t✐♦♥s✳

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✼✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥✱ ■■

Pr♦♣ ❚❤✐s ✐s ❛ ❞✐st❛♥❝❡ ❢✉♥❝t✐♦♥✱ ♥❛♠❡❧②✿ ✶✳ ✭❙②♠♠❡tr②✮ d(z✷, z✶) = d(z✶, z✷) ✷✳ ✭❚r✐❛♥❣✉❧❛r ✐♥❡q✉❛❧✐t②✮ d(z✶, z✸) ≤ d(z✶, z✷) + d(z✷, z✸) ✸✳ ✭❙❡♣❛r❛t✐♦♥✮ z✶ = z✷ = ⇒ d(z✶, z✷) > ✵

Pr♦♦❢✳

❙❡❡ ❬❇♦♥❛❤♦♥✱ ▲❡♠♠❛ ✷✳✶❪ ❢♦r ❛♥ ❡❧❡♠❡♥t❛r② ❛♥❞ ❞❡t❛✐❧❡❞ ♣r♦♦❢✳ ❋♦r s❡♣❛r❛t✐♦♥✿ ❖♥ t❤❡ r❡❣✐♦♥ U ⊂ H ❞❡✜♥❡❞ ❜② y ≤ ✷y✶✱ ✇❡ ❤❛✈❡ ·❤②♣

z

≥

✶ ✷y✶ ·❡✉❝

❤❡♥❝❡ ℓ❤②♣(c) ≥

✶ ✷y✶ ℓ❡✉❝(c) ❢♦r ❛❧❧ ❝✉r✈❡s c st❛②✐♥❣ ✐♥ U✳

▲❡t c ❜❡ ❛ ♣❛t❤ ❢r♦♠ z✶ t♦ z✷✳ ■❢ c ❧❡❛✈❡s U t❤❡♥ ℓ❤②♣(c) ≥

✶ ✷y✶ d❡✉❝(z✶, H − U) ≥ ✶ ✷y✶ y✶ = ✶ ✷✳

❍❡♥❝❡ d❤②♣(z✶, z✷) ≥ min( ✶

✷y✶ d❡✉❝(z✶, z✷), ✶ ✷) > ✵✳ ✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✽✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

Pr♦❥❡❝t✐♦♥ ♦♥ t❤❡ y✲❛①✐s

Pr♦♣ ▲❡t p : H → H ❜❡ ♣r♦❥❡❝t✐♦♥ ♦♥ t❤❡ y✲❛①✐s ✿ x + iy → iy✳ ▲❡t c : [a, b] → H ❜❡ ❛ ♣✇ C ✶ ♣❛t❤✳ ❚❤❡♥ ✶✳ ℓ(p ◦ c) ≤ ℓ(c) ✷✳ ■❢ ❡q✉❛❧✐t② t❤❡♥ c ✐s ✭❛♥ ✐♥❥❡❝t✐✈❡ ♣❛r❛♠❡tr✐③❛t✐♦♥ ♦❢✮ t❤❡ ✈❡rt✐❝❛❧ s❡❣♠❡♥t✳

Pr♦♦❢✳

❙❡❡ ❬❇♦♥❛❤♦♥✱ ▲❡♠♠❛ ✷✳✹❪✳

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✾✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❱❡rt✐❝❛❧ ❧✐♥❡s ❛r❡ s❤♦rt❡st ♣❛t❤s

❈♦r♦❧❧❛r②

▲❡t z✶, z✷ ❜❡ t✇♦ ♣♦✐♥ts ✐♥ H ♦♥ t❤❡ s❛♠❡ ✈❡rt✐❝❛❧ ❧✐♥❡✳ ❚❤❡♥ ✶✳ ❚❤❡ ✈❡rt✐❝❛❧ s❡❣♠❡♥t ✐s t❤❡ ✉♥✐q✉❡ s❤♦rt❡st ♣❛t❤ ❜❡t✇❡❡♥ z✶ ❛♥❞ z✷✳ ✷✳ d❤②♣(z✶, z✷) =

• ln y✷

y✶

• ✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧

✶✵✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

■s♦♠❡tr✐❡s

❉❡❢ ❆♥ ✐s♦♠❡tr② ❜❡t✇❡❡♥ t✇♦ ♠❡tr✐❝ s♣❛❝❡s X ❛♥❞ Y ✐s ❛ ❜✐❥❡❝t✐♦♥ f : X → Y ♣r❡s❡r✈✐♥❣ ❞✐st❛♥❝❡s✿ dY (f (z✶), f (z✷)) = dX(z✶, z✷) Pr♦♣s ◮ ❚❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s ✐s ❛♥ ✐s♦♠❡tr②✳ ◮ ✭❙❡❧❢✮✲■s♦♠❡tr✐❡s ♦❢ X ❢♦r♠ ❛ ❣r♦✉♣✱ ❞❡♥♦t❡❞ ❜② Isom(X)

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✶✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❋♦r ❛ C ✶ ♠❛♣ f : H → H ✐t ✐s ❡♥♦✉❣❤ t♦ ❝❤❡❝❦ t❤❛t f ♣r❡s❡r✈❡s t❤❡ ♥♦r♠ ♦❢ t❛♥❣❡♥t ✈❡❝t♦rs ✿ ❉❡❢ ❆♥ ❘✐❡♠❛♥♥✐❛♥ ✐s♦♠❡tr② ♦❢ H ✐s ❛ C ✶ ❜✐❥❡❝t✐♦♥ f : H → H s✉❝❤ t❤❛t ❢♦r ❡✈❡r② z ∈ H ❛♥❞ v ∈ C ❉zf (v)❤②♣

f (z) = v❤②♣ z

❚❤❡♥ f ♣r❡s❡r✈❡s t❤❡ ❤②♣❡r❜♦❧✐❝ ❧❡♥❣t❤ ♦❢ ♣❛t❤s ✿ ℓ(f ◦ c) = ℓ(c)✱ ❤❡♥❝❡ ✐s ❛♥ ✐s♦♠❡tr② ♦❢ H✳ ◆❇✿ ◆♦t❡ t❤❛t ✐t ❝❛♥ ❜❡ ♣r♦✈❡♥ t❤❛t✱ ❝♦♥✈❡rs❡❧②✱ ❛❧❧ ✐s♦♠❡tr✐❡s ♦❢ H ❛r❡ ❘✐❡♠❛♥♥✐❛♥ ✐s♦♠❡tr✐❡s✳

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✷✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❋✐rst ✐s♦♠❡tr✐❡s✱ ❛♥❞ ❍♦♠♦❣❡♥❡✐t②

❊①♦ ✶✳✷ ❈❤❡❝❦ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛♣s ❛r❡ ✐s♦♠❡tr✐❡s ♦❢ H✳ ◮ r❡✢❡❝t✐♦♥ ✐♥ t❤❡ y✲❛①✐s ✿ z → −z ◮ ❤♦r✐③♦♥t❛❧ tr❛♥s❧❛t✐♦♥s ✿ z → z + b✱ b ∈ R✳ ◮ ❤♦♠♦t❤❡t✐❡s ❝❡♥t❡r❡❞ ❛t ✵ ✿ z → az✱ a ∈ R∗ ❈♦♥❥✉❣❛t✐♥❣ ❜② ❤♦r✐③♦♥t❛❧ tr❛♥s❧❛t✐♦♥s ✇❡ ❣❡t ✿ ◮ r❡✢❡❝t✐♦♥s ❛❧♦♥❣ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ ◮ ❤♦♠♦t❤❡t✐❡s ✇✐t❤ ❝❡♥t❡r ♦♥ t❤❡ x✲❛①✐s

❈♦r♦❧❧❛r②

❚❤❡ ✉♣♣❡r ❤❛❧❢ ♣❧❛♥❡ H ✐s ❤♦♠♦❣❡♥❡♦✉s✱ ♥❛♠❡❧② Isom(H) ❛❝ts tr❛♥s✐t✐✈❡❧② ♦♥ H✳

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✸✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈❤❛♣t❡r ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✶✳✹ ●❡♦❞❡s✐❝s ✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✹✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥t❡♥t ♦❢ s❡❝t✐♦♥ ✶✳✷

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr②

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✶✺✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ ❙t❛♥❞❛r❞ ■♥✈❡rs✐♦♥

❘❡❢s ✿ ❬▲♦✉st❛✉ ❈❤❛♣ ✼❪✱ ❬❇❡❛r❞♦♥ ❈❤❛♣ ✸✱✹❪✱ ❬❘❛t❝❧✐✛❡ ❈❤❛♣ ✹❪ ❲❡ ✇♦r❦ ✐♥ t❤❡ ✉s✉❛❧ ❡✉❝❧✐❞❡❛♥ s♣❛❝❡ Rn, ✳ ❉❡❢ ❚❤❡ st❛♥❞❛r❞ ✐♥✈❡rs✐♦♥ ✐s t❤❡ ♠❛♣ σ : Rn − {✵} → Rn − {✵} ✏✐♥✈❡rs✐♥❣ t❤❡ ❞✐st❛♥❝❡s t♦ ✵✑✱ ♥❛♠❡❧② ✿ x → x x✷ σ ♥❛t✉r❛❧❧② ❡①t❡♥❞s t♦ ❛ s❡❧❢✲♠❛♣ ♦❢ Rn := Rn ∪ {∞} ❜②✿ σ(✵) = ∞ ❛♥❞ σ(∞) = ✵✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✶✻✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❘❡✢❡❝t✐♦♥✲❧✐❦❡ ♣r♦♣❡rt✐❡s

◮ σ ✜①❡s x ✐✛ x ✐s ✐♥ t❤❡ ✉♥✐t s♣❤❡r❡ S = Sn−✶ = {x ∈ Rn, x = ✶} ◮ σ ❡①❝❤❛♥❣❡s t❤❡ ✐♥t❡r✐♦r ❛♥❞ t❤❡ ❡①t❡r✐♦r ♦❢ S ◮ σ✷ = id ✭♥❛♠❡❧② σ ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥✮

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✶✼✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ t♦♣♦❧♦❣② ♦♥ Rn

• Rn := Rn ∪ {∞} ✐s ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ♦♥❡✲♣♦✐♥t ❝♦♠♣❛❝t✐✜❝❛t✐♦♥

✭♦r ❆❧❡①❛♥❞r♦✈ ❝♦♠♣❛❝t✐✜❝❛t✐♦♥✮ t♦♣♦❧♦❣②✱ ♥❛♠❡❧②✿ ♦♣❡♥ s❡ts ❛r❡ ◮ ❛❧❧ t❤❡ ♦♣❡♥ s✉❜s❡ts ♦❢ X = Rn ◮ t♦❣❡t❤❡r ✇✐t❤ ❛❧❧ s❡ts X \ K ∪ {∞} ✇❤❡r❡ K ✐s ❛ ❝♦♠♣❛❝t s✉❜s❡t ♦❢ X✳ ◆❇✿ ❆ s❡q✉❡♥❝❡ (xk)k ✐♥ Rn ❝♦♥✈❡r❣❡s t♦ ∞ ✐♥ Rn ✐✛ xk → ∞✳ Pr♦♣ Rn ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ s♣❤❡r❡ Sn

Pr♦♦❢✳

❆♥ ❡①♣❧✐❝✐t ❤♦♠❡♦♠♦r♣❤✐s♠ ✐s ❣✐✈❡♥ ❜② t❤❡ st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥ s : Sn − {en+✶} → Rn × {✵} ✐♥ Rn+✶✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✶✽✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ st❡r❡♦❣r❛♣❤✐❝ Pr♦❥❡❝t✐♦♥

❘❡❝❛❧❧✿ ❉❡❢ ❚❤❡ st❡r❡♦❣r❛♣❤✐❝ ♣r♦❥❡❝t✐♦♥ ✐s t❤❡ ♠❛♣ s s❡♥❞✐♥❣ x ∈ Sn − {en+✶} t♦ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ❢r♦♠ en+✶ t♦ x ✇✐t❤ t❤❡ ❤♦r✐③♦♥t❛❧ ❤②♣❡r♣❧❛♥❡ Rn × {✵} ⊂ Rn+✶✳ ❊①♦ ✶✳✸ ◮ ❋✐♥❞ t❤❡ ❛♥❛❧②t✐❝ ❡①♣r❡ss✐♦♥ ♦❢ s ◮ Pr♦✈❡ t❤❛t s ❞❡✜♥❡s ❛♥ ❤♦♠❡♦♠♦r♣❤✐s♠ Sn → Rn

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✶✾✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

• ❡♥❡r❛❧ ✐♥✈❡rs✐♦♥s

▲❡t S = S(a, R) = {x ∈ Rn, x − a = R} ❜❡ ❛♥② s♣❤❡r❡ ✐♥ Rn ❙❡♥❞✐♥❣ t❤❡ ✉♥✐t s♣❤❡r❡ t♦ S ❜② tr❛♥s❧❛t✐♥❣ ❛♥❞ ③♦♦♠✐♥❣✱ t❤❡ st❛♥❞❛r❞ ✐♥✈❡rs✐♦♥ ❜❡❝♦♠❡s ✿ ❉❡❢ ❚❤❡ ✐♥✈❡rs✐♦♥ ✐♥ S ✐s t❤❡ ♠❛♣ σS : Rn → Rn ❞❡✜♥❡❞ ❜② ✿ σS(x) = a +

R✷ x−a✷ (x − a) ✐❢ x = a, ∞✱

σS(a) = ∞✱ σS(∞) = a ❊①♦ ✶✳✹ Pr♦✈❡ t❤❛t σs ✐s ❛♥ ❤♦♠❡♦♠♦r♣❤✐s♠ ♦❢ Rn✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✵✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

• ❡♥❡r❛❧✐③❡❞ s♣❤❡r❡s

❍②♣❡r♣❧❛♥❡s ♦❢ Rn ♠❛② ❜❡ s❡❡♥ ❛s ✭❤②♣❡r✮s♣❤❡r❡s ♣❛ss✐♥❣ ❜② ∞ ✿ ❉❡❢ ❆ s♣❤❡r❡ ♦❢ Rn ✐s ❡✐t❤❡r✿ ◮ ❛ ✭❤②♣❡r✮s♣❤❡r❡ ♦❢ Rn ◮ ♦r P = P ∪ {∞}✱ ✇✐t❤ P ❛♥ ❤②♣❡r♣❧❛♥❡ ♦❢ Rn✳ ◆♦t❛t✐♦♥ ✿ ✐♥✈❡rs✐♦♥ σP ✐♥ P := ◮ ❡✉❝❧✐❞❡❛♥ r❡✢❡❝t✐♦♥ ✐♥ P ♦♥ Rn✱ ◮ ❡①t❡♥❞❡❞ t♦ Rn ❜② σP(∞) = ∞ ❊①♦ ✶✳✺ ✿ ❝❤❡❝❦ t❤✐s ✐s ❛♥ ❤♦♠❡♦✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✶✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

■♥✈❡rs✐♦♥s ♣r❡s❡r✈❡s s♣❤❡r❡s

Pr♦♣ ■♥✈❡rs✐♦♥s ♣r❡s❡r✈❡ s♣❤❡r❡s ♦❢ Rn✳ ◆❇✿ ❚❤❡ ✐♥✈❡rs✐♦♥ ✐♥ S(a, R) s❡♥❞s ❤②♣❡r♣❧❛♥❡s t♦ s♣❤❡r❡s ♣❛ss✐♥❣ ❜② a ❛♥❞ ❝♦♥✈❡rs❡❧②✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✷✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

◮ ◆♦r♠❛❧✐③❡ t♦ st❛♥❞❛r❞ ✐♥✈❡rs✐♦♥ ◮ ▲❡t S ❜❡ ❛ s♣❤❡r❡ ♦❢ Rn✳ ❙✉♣♣♦s❡ t❤❛t ∞ / ∈ S✱ t❤❛t ✐s t❤❛t S ✐s ❛♥ ♦r❞✐♥❛r② s♣❤❡r❡ S = S(a, R)✳ ▲❡t x ∈ Rn✱ x = ✵✳ ❉❡♥♦t❡ x∗ = σ(x) = x/x✷✳ ❚❤❡♥ x ∈ S ⇔ x − a✷ = x✷ − ✷a.x + a✷ = R✷ ⇔ ✶ − ✷a.

x x✷ + a✷−R✷ x✷

= ✵ ⇔ ✶ − ✷a.x∗ + (a✷ − R✷)x∗✷ = ✵ ✇❤✐❝❤ ✐s t❤❡ ❡q✉❛t✐♦♥ ❡✐t❤❡r ♦❢ ❛ s♣❤❡r❡ ✭✐❢ a = R✮ ♦r ♦❢ ❛ ❤②♣❡r♣❧❛♥❡ ✭✐❢ a = R✱ t❤❛t ✐s ✵ ∈ S✮✳ ◮ ❚❤❡ ❝❛s❡ ✇❤❡r❡ ∞ ∈ S ✐s s✐♠✐❧❛r✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✸✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

■♥✈❡rs✐♦♥s ❛r❡ ❝♦♥❢♦r♠❛❧

❘❡❝❛❧❧ ✿ ❉❡❢ ❆ ❞✐✛❡r❡♥t✐❛❜❧❡ ♠❛♣ f : Rn → Rn ✐s ❝♦♥❢♦r♠❛❧ ✐❢ ✐ts ❞✐✛❡r❡♥t✐❛❧ ❉xf : Rn → Rn ♣r❡s❡r✈❡s ❛♥❣❧❡s ✭t❤❛t ✐s✱ ✐s ❛ s✐♠✐❧❛r✐t②✮✳ Pr♦♣ ■♥✈❡rs✐♦♥s ❛r❡ ♦r✐❡♥t❛t✐♦♥ r❡✈❡rs✐♥❣ ❛♥❞ ❝♦♥❢♦r♠❛❧✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✹✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

Pr♦♦❢✳

◮ ❈♦♥❥✉❣❛t✐♥❣ ❜② ❛ tr❛♥s❧❛t✐♦♥ ❛♥❞ ❛♥ ❤♦♠♦t❤❡t② ✇❡ ❝❛♥ ♥♦r♠❛❧✐③❡ t♦ t❤❡ ❝❛s❡ ♦❢ t❤❡ st❛♥❞❛r❞ ✐♥✈❡rs✐♦♥ σ✳ ◮ ■♥ ❞✐♠❡♥s✐♦♥ ✷ t❤✐s ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t t❤✐s ✐s ❛♥ ❛♥t✐❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥✳ ◮ ■♥ ❣❡♥❡r❛❧ ❞✐♠❡♥s✐♦♥ ✿ ♦♥❡ ❝❛♥ s❤♦✇ t❤❛t ❉xσ = ✶ x✷ σx⊥ ✇❤❡r❡ σx⊥ ✐s t❤❡ r❡✢❡❝t✐♦♥ ✐♥ t❤❡ ❤②♣❡r♣❧❛♥❡ ♦rt❤♦❣♦♥❛❧ t♦ x✱ ❤❡♥❝❡ ❉xσ r❡✈❡rs❡ ♦r✐❡♥t❛t✐♦♥ ❛♥❞ ♣r❡s❡r✈❡s ❛♥❣❧❡s✳ ❙❡❡ ❬❘❛t❝❧✐✛❡ ❚❤♠ ✹✳✶✳✺❪ ❢♦r ❛ ❞❡t❛✐❧❡❞ ♣r♦♦❢ ✭♥♦t❡ t❤❛t t❤❡ ♠❛tr✐① ♦❢ σx⊥ ✐s I −

✷ x✷ xxt✮

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✺✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ ▼♦❡❜✐✉s ●r♦✉♣

❉❡❢ ❚❤❡ ▼♦❡❜✐✉s ●r♦✉♣ ♦❢ Rn ✐s t❤❡ ❣r♦✉♣ Mob( Rn) ❣❡♥❡r❛t❡❞ ❜② t❤❡ ✐♥✈❡rs✐♦♥s ✐♥ t❤❡ s♣❤❡r❡s ♦❢ Rn✳ ■♥❞✐✈✐❞✉❛❧ ❡❧❡♠❡♥ts ❛r❡ ❝❛❧❧❡❞ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s

❊①❛♠♣❧❡s

◮ ❊✉❝❧✐❞❡❛♥ ✐s♦♠❡tr✐❡s ♦❢ Rn ◮ ❤♦♠♦t❤❡t✐❡s ◆❇✿ ▼❛♣s f : Rn → Rn ❛r❡ ❡①t❡♥❞❡❞ t♦ Rn ❜② s❡tt✐♥❣ f (∞) = ∞✳ ❊①♦ ✶✳✻ ❈❤❡❝❦ t❤❛t t❤❡s❡ ❛r❡ ✐♥❞❡❡❞ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s ✭❢♦r ❤♦♠♦t❤❡t② ♦❢ ❝❡♥t❡r a✱ ❞❡❝♦♠♣♦s❡ ❛s ♣r♦❞✉❝t ♦❢ t✇♦ ✐♥✈❡rs✐♦♥s ✐♥ s♣❤❡r❡s ❝❡♥t❡r❡❞ ❛t a✮

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✻✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

Pr♦♣s ◮ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s ♣r❡s❡r✈❡s ▼♦❡❜✐✉s s♣❤❡r❡s ◮ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ❝♦♥❢♦r♠❛❧ ❲❡ ✇✐❧❧ ❞❡♥♦t❡ ❜② Mob+( Rn) t❤❡ ✐♥❞❡① ✷ s✉❜❣r♦✉♣ ♦❢ ♦r✐❡♥t❛t✐♦♥✲♣r❡s❡r✈✐♥❣ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s ✭r❡str✐❝t❡❞ ▼♦❡❜✐✉s ❣r♦✉♣✮✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✼✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

▲✐♥❡❛r ❛♥❞ ❛♥t✐❧✐♥❡❛r ❋r❛❝t✐♦♥❛❧ ▼❛♣s

■♥ ❞✐♠❡♥s✐♦♥ ✷✿ R✷ ✐❞❡♥t✐✜❡s ✇✐t❤ C✳ ˆ C := C ∪ {∞} ✐s ❝❛❧❧❡❞ t❤❡ ❘✐❡♠❛♥♥ ❙♣❤❡r❡✳ ❉❡❢ ❚❤❡ ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ❛r❡ t❤❡ ♠❛♣s C → C ♦❢ t❤❡ ❢♦r♠✿ z → az + b cz + d −d/c → ∞, ∞ → a/c ✇✐t❤ a, b, c, d ∈ C ❛♥❞ ad − bc = ✵✳ ❉❡❢ ❚❤❡ ❛♥t✐❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ❛r❡ t❤❡ ♠❛♣s C → C ♦❢ t❤❡ ❢♦r♠✿ z → az + b cz + d −d/c → ∞, ∞ → a/c ✇✐t❤ a, b, c, d ∈ C ❛♥❞ ad − bc = ✵✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✽✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

▼♦❡❜✐✉s ●r♦✉♣ ❛♥❞ ▲✐♥❡❛r ❋r❛❝t✐♦♥❛❧ ▼❛♣s

❚❤❡♦r❡♠

◮ Mob+( C) ✐s t❤❡ ❣r♦✉♣ ♦❢ ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s✳ ◮ Mob( C) ✐s t❤❡ ❣r♦✉♣ ♦❢ ❧✐♥❡❛r ♦r ❛♥t✐❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✷✾✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

■♥❣r❡❞✐❡♥ts ♦❢ t❤❡ ♣r♦♦❢

◮ ❚❛❦✐♥❣ ♣r♦❞✉❝ts ♦❢

◮ ❚r❛♥s❧❛t✐♦♥ ✿ z → z + b✱ b ∈ C ◮ ❙✐♠✐❧❛r✐t✐❡s ✿ z → az✱ a ∈ C∗ ◮ ❘❡✢❡❝t✐♦♥ ✐♥ t❤❡ x✲❛①✐s ✿ z → z ◮ ❙t❛♥❞❛r❞ ✐♥✈❡rs✐♦♥ ✿ σ : z → ✶/z

t❤❛t ❛r❡ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s✱ ✇❡ ❝❛♥ s❡❡ t❤❛t ❛❧❧ ❧✐♥❡❛r ❛♥❞ ❛♥t✐❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ❛r❡ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s ✭✉s✐♥❣ t❤❛t az+b

cz+d = a c − ad−bc c(cz+d)✮

◮ ▲✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ❢♦r♠ ❛ ❣r♦✉♣ ✭❜❡tt❡r s❡❡♥ ✐❞❡♥t✐❢②✐♥❣ t❤❡♠ ✇✐t❤ PGL✷(C) s❡❡ ♥❡①t s❡❝t✐♦♥✮ ◮ ▲✐♥❡❛r ♦r ❛♥t✐❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ❢♦r♠ ❛ ❣r♦✉♣ ✭s✐♥❝❡ ❛♥t✐❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ❛r❡ ♦❢ t❤❡ ❢♦r♠ f (z) ✇✐t❤ f ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧✱ ❛♥❞ f ❝♦♠♠✉t❡s ✇✐t❤ ❝♦♥❥✉❣❛❝② ✿ f (z) = f (z)✮ ◮ ❆❧❧ ✐♥✈❡rs✐♦♥s ❛r❡ ❛♥t✐❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ✿ t❤❡② ❛r❡ ❝♦♥❥✉❣❛t❡❞ t♦ ❡✐t❤❡r t❤❡ r❡✢❡❝t✐♦♥ ✐♥ t❤❡ x✲❛①✐s ♦r t❤❡ st❛♥❞❛r❞ ✐♥✈❡rs✐♦♥ ❜② s♦♠❡ z → eiθz + b ✭r♦t❛t✐♦♥✱ t❤❡♥ tr❛♥s❧❛t✐♦♥✮✳ ❊①♦ ✶✳✼ ❈❤❡❝❦ ❞❡t❛✐❧s ❛♥❞ ✜♥✐s❤ t❤❡ ♣r♦♦❢

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✸✵✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥t❡♥t ♦❢ s❡❝t✐♦♥ ✶✳✷

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✶ ■♥✈❡rs✐♦♥s ❛♥❞ ▼ö❜✐✉s ●r♦✉♣ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr②

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✶✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

Pr♦❥❡❝t✐✈❡s s♣❛❝❡s

▲❡t K ❜❡ ❛♥② ✜❡❧❞✳ ❈♦♥s✐❞❡r ❛ ✈❡❝t♦r s♣❛❝❡ V ♦✈❡r K✳ ❉❡❢ ❚❤❡ ♣r♦❥❡❝t✐✈❡ s♣❛❝❡ ♦❢ V ✐s t❤❡ s❡t P(V ) ♦❢ ❧✐♥❡s ✐♥ V ✳ ◮ ❲❤❡♥ V = Kn+✶✱ t❤❡ ♣r♦❥❡❝t✐✈❡ s♣❛❝❡ P(Kn+✶) ✐s ❛❧s♦ ❞❡♥♦t❡❞ PnK✱ Pn(K) ♦r KPn✳ ◮ ❋♦r x ∈ V − {✵}✱ ✇❡ ❞❡♥♦t❡ [x] = Kx t❤❡ ❧✐♥❡ ❝♦♥t❛✐♥✐♥❣ x✳ ◮ P(V ) ✐❞❡♥t✐✜❡s t♦ t❤❡ q✉♦t✐❡♥t s❡t (V − ✵)/ ∼ ✇❤❡r❡ x ∼ y ✐❢ [x] = [y] ✐✛ ∃λ ∈ K∗, y = λx ✭❝♦❧✐♥❡❛r✐t②✮ ◮ ❲❤❡♥ K ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ ✜❡❧❞ ✭❡❣ K = R ♦r C✮✱ P(V ) ✐s ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ q✉♦t✐❡♥t t♦♣♦❧♦❣②

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✷✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ Pr♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣

◮ ❉❡❢ ❆ ♠❛♣ P(V ) → P(W ) ✐s ♣r♦❥❡❝t✐✈❡ ✐❢ ✐t ✐s t❤❡ ♠❛♣ f : [x] → [f (x)] ✐♥❞✉❝❡❞ ❜② ❛♥ ✐♥❥❡❝t✐✈❡ ❧✐♥❡❛r ♠❛♣ f : V → W ✳ ❊①♦ ✶✳✽ ❝❤❡❝❦ t❤❛t

◮ f ✐s ❛❧✇❛②s ✇❡❧❧✲❞❡✜♥❡❞ ◮ f ◦ g = f ◦ g

◮ GL(V ) ❛❝ts ♦♥ P(V ) ❜② g · [x] = [g(x)] ✭♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥s✮ ◮ ❚❤❡ ❦❡r♥❡❧ ♦❢ t❤✐s ❛❝t✐♦♥ Ψ : GL(V ) → Bij(P(V )) ✐s t❤❡ s✉❜❣r♦✉♣ K∗ id ♦❢ ❤♦♠♦t❤❡t✐❡s ◮ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ✐s PGL(V ) := GL(V )/K∗ id ◮ PGL(Kn) ✐❞❡♥t✐✜❡s ✇✐t❤ PGLn(K) := GLn(K)/K∗In ✭❛❧s♦ ❞❡♥♦t❡❞ ❜② PGL(n, K)✮✳ ◮ ❋♦r G < GL(V ) t❤❡ ♣r♦❥❡❝t✐✈❡ ❣r♦✉♣ ♦❢ G ✐s✿ PG := G/G∩K∗ id ❊①❛♠♣❧❡✿ PSL(V ) ✐s t❤❡ ♣r♦❥❡❝t✐✈❡ ❣r♦✉♣ ♦❢ SL(V ) ✭❞ét❡r♠✐♥❛♥t ✶ ❡♥❞♦♠♦r♣❤✐s♠s✮

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✸✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ Pr♦❥❡❝t✐✈❡ ▲✐♥❡

◮ ❲❤❡♥ dim V = ✷✱ P(V ) ✐s ❝❛❧❧❡❞ ❛ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡✳ ◮ ❚❤❡ ❝❤♦✐❝❡ ♦❢ ❛ ❜❛s✐s ♦❢ V ②✐❡❧❞s ❝❛♥♦♥✐❝❛❧ ✐❞❡♥t✐✜❝❛t✐♦♥ ✇✐t❤ t❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡ P✶K = P(K✷)✳ ◮ ❋♦r ♥♦♥③❡r♦ (x✶, x✷) ∈ K✷✱ ✇❡ ❛❜❜r❡✈✐❛t❡ [(x✶, x✷)] t♦ [x✶ : x✷]✳ ❚❤✐s ✐s ❝❛❧❧❡❞ ❤♦♠♦❣❡♥❡♦✉s ❝♦♦r❞✐♥❛t❡s✳ ◮ ■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ P✶K ✇✐t❤ K := K ∪ {∞}✿ ❚❤❡ ❝❛♥♦♥✐❝❛❧ ❛✣♥❡ ❝❤❛rt ♦❢ P✶K ✐s t❤❡ ♠❛♣ ψ : P✶K → K ∪ {∞} ❞❡✜♥❡❞ ❜② ψ([x✶ : x✷]) = x✶/x✷ ✐❢ x✷ = ✵ ψ([x✶ : ✵]) = ∞ ■t ✐s ❛ ❜✐❥❡❝t✐♦♥✱ ✇❤♦s❡ ✐♥✈❡rs❡ s❡♥❞s x ∈ K t♦ [x, ✶] ❛♥❞ ∞ t♦ [✶ : ✵] ✭❡①♦✮✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✹✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

PGL✷(K) ❛♥❞ ▲✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s

▲✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❛❝t✐♦♥ ♦❢ PGL✷(K) ♦♥ t❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡ P✶K✱ ✉♥❞❡r t❤❡ ❝❛♥♦♥✐❝❛❧ ✐❞❡♥t✐✜❝❛t✐♦♥ P✶K ≃ K ∪ {∞} ✭✈✐❛ t❤❡ ❛✣♥❡ ❝❤❛rt✮✿ ■♥❞❡❡❞ ✿ ▲❡t g = a b c d

• ∈ GL✷(K)✳

❲❡ ❞❡♥♦t❡ ϕg : x → g · x t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❛♣ K → K✳ ❚❤❡♥ ❋♦r x ∈ K, ϕg(x) = g · [x : ✶] = [ax + b : cx + d] = [ ax+b

cx+d : ✶] = ax+b cx+d

✐❢ cx + d = ✵ [✶ : ✵] = ∞ ✐❢ cx + d = ✵ ❆♥❞ ϕg(∞) = g · [✶ : ✵] = [a : c] = a/c ❚❤❡ ♠❛♣ g → ϕg ❢r♦♠ GL✷(K) t♦ Bij( K) ✐s ❛ ❤♦♠♦♠♦r♣❤✐s♠✳ ❈♦r♦ ▲✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s K → K ❢♦r♠ ❛ ❣r♦✉♣ ✐s♦♠♦r♣❤✐❝ t♦ PGL✷(K)✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✺✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈r♦ss✲r❛t✐♦ ■

▲❡t P(V ) ❜❡ ❛ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡✳ Pr♦♣ PGL(V ) ❛❝ts ✶✲tr❛♥s✐t✐✈❡❧② ♦♥ t❤❡ s❡t P(V )✸∗ ♦❢ tr✐♣❧❡s ♦❢ ❞✐st✐♥❝t ♣♦✐♥ts ✭❡①♦✮ ❉❡❢ ▲❡t p, x, y, q ❜❡ ❢♦✉r ❞✐st✐♥❝t ♣♦✐♥ts ✐♥ P(V ) ❚❤❡ ❝r♦ss✲r❛t✐♦ [p, x, y, q] ♦❢ (p, x, y, q) ✐s t❤❡ ✉♥✐q✉❡ a ∈ K s✉❝❤ t❤❛t (p, x, y, q) ❝❛♥ ❜❡ s❡♥t t♦ (✵, ✶, a, ∞) ❜② ❛ ♣r♦❥❡❝t✐✈❡ ✐s♦♠♦r♣❤✐s♠ P(V ) → P✶K ≃ K ∪ {∞}✳ ❊①♦ ✶✳✾ ❈❤❡❝❦ t❤✐s ✐s ✇❡❧❧✲❞❡✜♥❡❞ Pr♦♣ PGL(V ) ♣r❡s❡r✈❡s t❤❡ ❝r♦ss✲r❛t✐♦

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✻✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈r♦ss✲r❛t✐♦ ■■

Pr♦♣ ▲❡t p, x, y, q ❜❡ ❢♦✉r ❞✐st✐♥❝t ♣♦✐♥ts ✐♥ K✳ ❚❤❡♥ [p, x, y, q] = y − p x − p .x − q y − q

Pr♦♦❢✳

❉❡♥♦t❡ a = y−p

x−p. x−q y−q✳ ❚❤❡ ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣ t → t−p x−p. x−q t−q ✐s ❛

♣r♦❥❡❝t✐✈❡ ✐s♦♠♦r♣❤✐s♠ K → K s❡♥❞✐♥❣ (p, x, y, q) t♦ (✵, ✶, a, ∞)✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✼✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈r♦ss✲r❛t✐♦ ■■■

❊①♦ ✶✳✶✵ ❚❤❡ ♦r❞❡r ♦❢ t❤❡ ♣♦✐♥ts ✐s ✐♠♣♦rt❛♥t✳ ❈❤❡❝❦ t❤❛t ◮ [p, y, x, q] = [p, x, y, q]−✶❀ ◮ [q, x, y, p] = [p, x, y, q]−✶❀ ◮ [x, p, y, q] = ✶ − [p, x, y, q] Pr♦♣ ◮ [p, x, z, q] = [p, x, y, q].[p, y, z, q] ✭❡①♦✮

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✽✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

• ❡♦♠❡tr✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣❧❡① ❝r♦ss✲r❛t✐♦

K = C✱ P✶C ≃ C ∪ {∞}✳ Pr♦♣ ✶✳ ❚❤r❡❡ ❞✐st✐♥❝t ♣♦✐♥ts z✱ α✱ β ✐♥ C ❛r❡ ❛❧✐❣♥❡❞ ⇔ [z, α, ∞, β] ∈ R✳ Pr♦♦❢ ✿ ⇔ t❤❡ ❛♥❣❧❡ ❛t z ✐s ✐♥ πZ ⇔ z−α

z−β ∈ R

✷✳ ❋♦✉r ❞✐st✐♥❝t ♣♦✐♥ts z✱ α✱ω✱ β ✐♥ C ❛r❡ ❝♦❝②❝❧✐❝ ⇔ [z, α, ω, β] ∈ R✳

✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✷✳✷ ❚❤❡ ♣r♦❥❡❝t✐✈❡ ❧✐♥❡❛r ❣r♦✉♣ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ❣❡♦♠❡tr② ✸✾✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈❤❛♣t❡r ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✶✳✹ ●❡♦❞❡s✐❝s ✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s

✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✹✵✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❉❡✜♥✐t✐♦♥

◮ ❚❤❡ P♦✐♥❝❛ré ❞✐s❦ ✐s t❤❡ ✉♥✐t ❞✐s❦ B = {z ∈ C, |z| < ✶} ◮ ❚❤❡ ✭❤②♣❡r❜♦❧✐❝✮ ♥♦r♠ ❛t z ∈ B ♦❢ ❛ ✈❡❝t♦r v ∈ R✷ ≃ C ✐s vB

z :=

✷ ✶ − |z|✷ v❡✉❝ ◮ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❛ C ✶✲❝✉r✈❡ c : [a, b] → H ✐s ℓB(c) := b

a

c′(t)B

c(t) ❞t

◮ ❚❤❡ ✭❤②♣❡r❜♦❧✐❝✮ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts z✶, z✷ ∈ B ✐s dB(z✶, z✷) := inf{ℓB(c)} ✇❤❡r❡ t❤❡ ✐♥❢ ✐s t❛❦❡♥ ♦✈❡r ❛❧❧ ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ C ✶ ♣❛t❤s c ❢r♦♠ z✶ t♦ z✷ ✐♥ B✳

✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✹✶✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❊①♦ ✶✳✶✶ ❈♦♠♣✉t❡ t❤❡ ❤②♣❡r❜♦❧✐❝ ❧❡♥❣t❤ ♦❢ ✶✳ t❤❡ s❡❣♠❡♥t ❜❡t✇❡❡♥ ✵ ❛♥❞ r ∈ [✵, ✶[✳ ✷✳ t❤❡ ❡✉❝❧✐❞❡❛♥ ❝✐r❝❧❡ ♦❢ ❝❡♥t❡r ✵ ❛♥❞ r❛❞✐✉s r ∈ [✵, ✶[✳ ❊①♦ ✶✳✶✷ Pr♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ✿ ◮ dB ✐s ❛ ♠❡tr✐❝ ♦♥ B ✭❢♦r s❡♣❛r❛t✐♦♥✱ ♥♦t❡ t❤❛t dB ≥ ✷d❡✉❝✮✳ ◮ ❘♦t❛t✐♦♥s ❝❡♥t❡r❡❞ ❛t ✵ ❛r❡ ✐s♦♠❡tr✐❡s ♦❢ B ◮ ❘❡✢❡❝t✐♦♥s ✐♥ ❧✐♥❡s ♣❛ss✐♥❣ ❜② ✵ ❛r❡ ✐s♦♠❡tr✐❡s ♦❢ B ❊①♦ ✶✳✶✸ ❋✐♥❞ ❛ ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣ s❡♥❞✐♥❣ H t♦ B ✭❤✐♥t✿ ✐ts ✐s ❡♥♦✉❣❤ t♦ s❡♥❞ (✵, ✶, ∞) t♦ (−✶, −i, ✶)✮✳

✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✹✷✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ ❝♦rr❡s♣♦♥❞❛♥❝❡ ❜❡t✇❡❡♥ H ❛♥❞ B

❚❤❡♦r❡♠

❚❤❡ ♠❛♣ Φ : H → B z →

z−i z+i

✐s ❛♥ ✐s♦♠❡tr②✳ ◮ Φ ✐s ❛ ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣✱ ✐♥ ♣❛rt✐❝✉❧❛r Φ ✐s ♦r✐❡♥t❛t✐♦♥✲♣r❡s❡r✈✐♥❣✱ ❝♦♥❢♦r♠❛❧ ❛♥❞ ♣r❡s❡r✈❡s ❝✐r❝❧❡s ♦❢ C ◮ ♥♦t❡ t❤❛t Φ(i) = ✵ ◮ Φ s❡♥❞s ✵ t♦ −✶✱ ∞ t♦ ✶ ❛♥❞ ✶ t♦ −i✱ ◮ ❤❡♥❝❡ Φ s❡♥❞s t❤❡ ❝✐r❝❧❡ R ∪ {∞} t♦ t❤❡ ✉♥✐t ❝✐r❝❧❡✱ t❤❡♥ H t♦ B✳

✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✹✸✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

Pr♦♦❢

❲❡ ❤❛✈❡ ❉zΦ(v) = Φ′(z)v ❛♥❞ Φ′(z) =

✷i (z+i)✷

❉zΦ(v)❤②♣

Φ(z)

=

✷ ✶−|Φ(z)|✷ |Φ′(z)v|

=

✷ ✶− |z−i|✷

|z+i|✷

✷ |z+i|✷ |v|

=

✹ |z+i|✷−|z−i|✷ |v|

=

✶ Im(z) |v|

= v❤②♣

z

✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✹✹✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈❤❛♣t❡r ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✶✳✹ ●❡♦❞❡s✐❝s ✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s

✶✳✹ ●❡♦❞❡s✐❝s ✹✺✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

• ❡♦❞❡s✐❝s ♦❢ t❤❡ ❤②♣❡r❜♦❧✐❝ ♣❧❛♥❡

❚❤❡♦r❡♠

❚❤❡ ❣❡♦❞❡s✐❝s ♦❢ H ❛r❡ t❤❡ ❝✐r❝✉❧❛r ❛r❝s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ❜♦✉♥❞❛r② ✭t❤❡ x✲❛①✐s✮ ❛♥❞ str❛✐❣❤t ✈❡rt✐❝❛❧ r❛②s✳

❚❤❡♦r❡♠

❚❤❡ ❣❡♦❞❡s✐❝s ♦❢ B ❛r❡ t❤❡ ❝✐r❝❧❡ ❛r❝s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ❜♦✉♥❞❛r② ✭t❤❡ ✉♥✐t ❝✐r❝❧❡✮✳

✶✳✹ ●❡♦❞❡s✐❝s ✹✻✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈❤❛♣t❡r ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✶✳✹ ●❡♦❞❡s✐❝s ✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✹✼✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥t❡♥t ♦❢ s❡❝t✐♦♥ ✶✳✺

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✶✳✺✳✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✐♥✜♥✐t② ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✹✽✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

PGL✷(R) ❛❝t✐♥❣ ♦♥ C

◮ ❚❤❡ ✐♥❝❧✉s✐♦♥ GL✷(R) ⊂ GL✷(C) ✐♥❞✉❝❡s ❛♥ ✐♥❝❧✉s✐♦♥ PGL✷(R) ⊂ PGL✷(C) ✭❝❤❡❝❦✮✳ ❍❡♥❝❡ PGL✷(R) ❛❝ts ♦♥ P✶C ≃ C✳ ◮ ❚❤❡ ✐♥❝❧✉s✐♦♥ R✷ ⊂ C✷ ✐♥❞✉❝❡s ❛♥ ✐♥❝❧✉s✐♦♥ P✶R ⊂ P✶C✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t❛❦✐♥❣ st❛♥❞❛r❞ ❛✣♥❡ ❝❤❛rts t♦ t❤❡ ♥❛t✉r❛❧ ✐♥❝❧✉s✐♦♥ R ⊂ C ✭❝❤❡❝❦✮✳ ◮ PGL✷(R) ❛❝t✐♦♥ ♦♥ C ❝❧❡❛r❧② ♣r❡s❡r✈❡s R✳ ❊①♦ ✶✳✶✹ ❋✐♥❞ t❤❡ st❛❜✐❧✐③❡r S

R ♦❢

R ✐♥ PGL✷(C)✳ ✳✳✳ ❆♥s✇❡r✿

✷

✳ Pr♦♦❢✿ ✐❢

✷

♣r❡s❡r✈❡s ✱ t❤❡♥ ✵ ✶

✸ ✱

❤❡♥❝❡ t❤❡r❡ ❡①✐sts

✷

s❡♥❞✐♥❣ ✵ ✶ t♦ ✵ ✶ ✐♥ ✱ ❤❡♥❝❡ ✐♥ ✳ ❇② ✶✲tr❛♥s✐t✐✈✐t② ♦❢

✷

♦♥

✸ ✱

✇❡ t❤❡♥ ❤❛✈❡ ✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✹✾✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

PSL✷(R) ♣r❡s❡r✈❡s H

Pr♦♣ ❚❤❡ ❛❝t✐♦♥ ♦❢ PSL✷(R) ♦♥ C ✭❜② ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s z → az+b

cz+d ✮ ♣r❡s❡r✈❡s t❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ H✳

❉✐r❡❝t ♣r♦♦❢✳

▲❡t g = a b c d

• ∈ SL✷(R)✱ ❛♥❞ z ∈ H t❤❡♥ ❛ ❝♦♠♣✉t❛t✐♦♥ ❣✐✈❡s

Im(ϕg(z)) =

Im(z) |cz+d|✷ > ✵✳

▼♦r❡ ❝♦♥❝❡♣t✉❛❧ ♣r♦♦❢✿ ✉s✐♥❣ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ R ../..

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✺✵✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

▲✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ♣r❡s❡r✈✐♥❣ H

❈♦♥s✐❞❡r t❤❡ ♥❛t✉r❛❧ ♦r✐❡♥t❛t✐♦♥ ♦♥ t❤❡ ❝✐r❝❧❡ R ✭= ♦r❞❡r ♦♥ R✮✳ Pr♦♣ ▲❡t ϕ ∈ PGL✷(C)✳ ❚❋❆❊✿ ✭✐✮ ϕ ♣r❡s❡r✈❡s H ✭✐✐✮ ϕ ♣r❡s❡r✈❡s t❤❡ ♦r✐❡♥t❡❞ ❝✐r❝❧❡ R ✭✐✐✐✮ ϕ ∈ PGL+

✷ (R) = PSL✷(R)

■♥ ♦t❤❡r ✇♦r❞s✿ StabPGL✷(C)(H) = Stab+

PGL✷(C)(

R) = PSL✷(R) ❲❤❡r❡ ◮ Stab+

PGL✷(C)(

R) ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ ❡❧❡♠❡♥ts ♦❢ StabPGL✷(C)( R) = PGL✷(R) ♣r❡s❡r✈✐♥❣ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ R✳ ◮ GL+

✷ (R) ✐s t❤❡ s✉❜❣r♦✉♣ ♦❢ ❡❧❡♠❡♥ts ♦❢ GL✷(R) ✇✐t❤ ♣♦s✐t✐✈❡

❞❡t❡r♠✐♥❛♥t✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✺✶✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

Pr♦♦❢✳

◮ PGL+

✷ (R) = PSL✷(R) ✿ ■❢ det(g) > ✵✱ t❛❦✐♥❣ ✶

√

det(g)g ✇❡ ❣❡t

❛ r❡♣r❡s❡♥t❛♥t ♦❢ ❞❡t ✶✳ ◮ ✭✐✮⇔✭✐✐✮✿ ❛s ϕ ✐s ❛♥ ❤♦♠❡♦ ❛♥❞ ♣r❡s❡r✈❡s t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦❢ C ◮ ✭✐✐✐✮⇒✭✐✐✮✿ PSL✷(R) ✐s ❣❡♥❡r❛t❡❞ ❜② z → a✷z + b✱ a ∈ R∗✱ b ∈ R ❛♥❞ z → −✶/z ✭❡①♦✮✱ ✇❤✐❝❤ ❛❧❧ ♣r❡s❡r✈❡ t❤❡ ♦r✐❡♥t❛t✐♦♥ ♦♥ R✳ ◮ ✭✐✐✮⇒ ✭✐✐✐✮✿

◮ PGL+

✷ (R) ⊂ Stab+ PGL✷(C)(

R) ◮ PGL+

✷ (R) ❛♥❞ Stab+ PGL✷(C)(

R) ❜♦t❤ ❤❛✈❡ ✐♥❞❡① ✷ ✐♥ PGL✷(R) ◮ ❤❡♥❝❡ t❤❡② ❛r❡ ❡q✉❛❧

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✺✷✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

PSL✷(R) ❛♥❞ ■s♦♠❡tr✐❡s ♦❢ H

▲❡♠♠❛

PSL✷(R) ❛❝ts ❜② ✐s♦♠❡tr✐❡s ♦♥ H

Pr♦♦❢✳

◮ ❉✐r❡❝t Pr♦♦❢✿ ❝♦♠♣✉t❡ ϕ′

g(z) ❛♥❞ ❞✐r❡❝t❧② ❝❤❡❝❦ t❤❛t ❉ϕg

♣r❡s❡r✈❡s t❤❡ ❤②♣❡r❜♦❧✐❝ ♥♦r♠ ♦❢ ✈❡❝t♦rs ◮ ❆♥♦t❤❡r Pr♦♦❢✿

◮ ❚❤❡ ✐♥✈❡rs✐♦♥s ✐♥ ❝✐r❝❧❡s ♣❛ss✐♥❣ ❜② i ❛♥❞ ♦rt❤♦❣♦♥❛❧ t♦ R ❛r❡ ❝♦♥❥✉❣❛t❡❞ ❜② Φ : H → B t♦ r❡✢❡❝t✐♦♥s t❤r♦✉❣❤ ❧✐♥❡s t❤r♦✉❣❤ ✵✱ ❤❡♥❝❡ ❛r❡ ❤②♣❡r❜♦❧✐❝ ✐s♦♠❡tr✐❡s✳ ◮ ❤❡♥❝❡ z → −✶/z ✐s ❛♥ ✐s♦♠❡tr② ♦❢ H✳ ◮ PSL✷(R) ✐s ❣❡♥❡r❛t❡❞ ❜② z → az + b ✇✐t❤ a, b ∈ R ❛♥❞ a > ✵✱ t❤❛t ❛r❡ ✐s♦♠❡tr✐❡s ♦❢ H✱ ❛♥❞ z → −✶/z✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✺✸✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥t❡♥t ♦❢ s❡❝t✐♦♥ ✶✳✺

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✶✳✺✳✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✐♥✜♥✐t② ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✺✹✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣

❉❡♥♦t❡ ❜② Isom+(H) t❤❡ ❣r♦✉♣ ♦❢ ♦r✐❡♥t❛t✐♦♥✲♣r❡s❡r✈✐♥❣ ✐s♦♠❡tr✐❡s ♦❢ H✳

❚❤❡♦r❡♠

Isom+(H) = PSL✷(R)

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✺✺✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❖r✐❡♥t❛t✐♦♥✲r❡✈❡rs✐♥❣ ✐s♦♠❡tr✐❡s

❊①♦ ✶✳✶✺ ❙❤♦✇ t❤❛t ♦r✐❡♥t❛t✐♦♥✲r❡✈❡rs✐♥❣ ✐s♦♠❡tr✐❡s ♦❢ H ❛r❡ t❤❡ ♠❛♣ ♦❢ t❤❡ ❢♦r♠ z → ϕ(−z)✱ ✇✐t❤ ϕ ∈ Isom+(H)✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✺✻✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ st❛❜✐❧✐③❡r ♦❢ i

Pr♦♣ ❚❤❡ st❛❜✐❧✐③❡r ♦❢ i ✐♥ PSL✷(R) ✐s PSO✷(R)✳ ✭❡①♦✮

❈♦r♦❧❧❛r②

H ≃ PSL✷(R)/ PSO✷(R)✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✺✼✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥t❡♥t ♦❢ s❡❝t✐♦♥ ✶✳✺

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✶✳✺✳✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✐♥✜♥✐t② ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✺✽✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

■s♦♠❡tr✐❡s ♦❢ B

❚❤❡♦r❡♠

❚❤❡ ♦r✐❡♥t❛t✐♦♥✲♣r❡s❡r✈✐♥❣ ✐s♦♠❡tr✐❡s ♦❢ B ❛r❡ t❤❡ r❡str✐❝t✐♦♥ t♦ B ♦❢ t❤❡ ❧✐♥❡❛r ❢r❛❝t✐♦♥❛❧ ♠❛♣s ϕ ♦❢ t❤❡ ❢♦r♠ ϕ(z) = αz + β βz + α ✇✐t❤ α, β ∈ C ❛♥❞ |α|✷ − |β|✷ = ✶✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✺✾✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

▲✐♥❦ ✇✐t❤ PU(✶, ✶)

◮ ❘❡❝❛❧❧ t❤❛t U(✶, ✶) < GL✷(C) ✐s t❤❡ s✉❜❣r♦✉♣ ♣r❡s❡r✈✐♥❣ t❤❡ st❛♥❞❛r❞ ❤❡r♠✐t✐❛♥ ❢♦r♠ h(z, w) = z✶w✶ − z✷w✷ ♦❢ s✐❣♥❛t✉r❡ (✶, ✶) ◮ g ∈ U(✶, ✶) ✐✛ g = α β β α

• ✇✐t❤ α, β ∈ C ❛♥❞

|α|✷ − |β|✷ = ✶✳ ◮ U(✶, ✶) ♣r❡s❡r✈❡s t❤❡ ♥❡❣❛t✐✈❡ ❝♦♥❡ ♦❢ h✿ V− = {z ∈ C✷, h(z, z) = |z✶|✷ − |z✷|✷ < ✵}✱ ❤❡♥❝❡ PU(✶, ✶) ♣r❡s❡r✈❡s P(V−) ⊂ P✶C ◮ ■♥ t❤❡ st❛♥❞❛r❞ ❛✣♥❡ ❝❤❛rt z✷ = ✶✱ P(V−) ✐❞❡♥t✐✜❡s t♦ B✱ ❤❡♥❝❡ PU(✶, ✶) ♣r❡s❡r✈❡s B✳

❚❤❡♦r❡♠

Isom+(B) = PU(✶, ✶)

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✻✵✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥t❡♥t ♦❢ s❡❝t✐♦♥ ✶✳✺

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✶✳✺✳✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✐♥✜♥✐t② ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✐♥✜♥✐t② ✻✶✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❉❡✜♥✐t✐♦♥

◮ ❚❤❡ ❜♦✉♥❞❛r② ∂B ♦❢ B ✐s t❤❡ ✉♥✐t ❝✐r❝❧❡ ∂B = S✶ = {z ∈ C, |z| = ✶✳ ❲❡ ❞❡♥♦t❡ B = B ∪ ∂B t❤❡ ❝❧♦s✉r❡ ♦❢ B ✐♥ C✳ ◮ ❚❤❡ ❜♦✉♥❞❛r② ∂H ♦❢ H ✐s ∂H = R ∪ {∞}✳ ❲❡ ❞❡♥♦t❡ H = H ∪ ∂H t❤❡ ❝❧♦s✉r❡ ♦❢ H ✐♥ C✳ ❚❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❜♦✉♥❞❛r② ❝✐r❝❧❡ ❛r❡ ❝❛❧❧❡❞ ✐❞❡❛❧ ♣♦✐♥ts✳ ◆♦t❡ t❤❛t✱ s✐♥❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s♦♠❡tr✐❡s ❛r❡ r❡str✐❝t✐♦♥s ♦❢ ▼♦❡❜✐✉s tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ C✱ ✇❡ ❤❛✈❡ ◮ Φ : H → B ❡①t❡♥❞s t♦ ❛♥ ❤♦♠❡♦♠♦r♣❤✐s♠ H → B ◮ ❚❤❡ ✐s♦♠❡tr✐❡s ♦❢ B ❡①t❡♥❞ t♦ ❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ B ◮ ❚❤❡ ✐s♦♠❡tr✐❡s ♦❢ H ❡①t❡♥❞ t♦ ❤♦♠❡♦♠♦r♣❤✐s♠s ♦❢ H

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✐♥✜♥✐t② ✻✷✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥t❡♥t ♦❢ s❡❝t✐♦♥ ✶✳✺

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✶ PSL✷(R) ❛❝ts ♦♥ H ❜② ✐s♦♠❡tr✐❡s ✶✳✺✳✷ ❚❤❡ ❢✉❧❧ ✐s♦♠❡tr② ❣r♦✉♣ ✶✳✺✳✸ ■s♦♠❡tr✐❡s ♦❢ B ✶✳✺✳✹ ❊①t❡♥s✐♦♥ t♦ t❤❡ ❜♦✉♥❞❛r② ❛t ✐♥✜♥✐t② ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s ✻✸✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❉❡❢ ▲❡t g ∈ PSL✷(R)✳ g ✐s ❝❛❧❧❡❞ ◮ ❡❧❧✐♣t✐❝ ✐❢ g ❤❛s ❛ ✜①❡❞ ♣♦✐♥t ✐♥ H ❀ ◮ ♣❛r❛❜♦❧✐❝ ✐❢ g ❤❛s ♥♦ ✜①❡❞ ♣♦✐♥t ✐♥ H✱ ❛♥❞ ❡①❛❝t❧② ♦♥❡ ✐♥ ∂H ◮ ❤②♣❡r❜♦❧✐❝ ✐❢ g ❤❛s ♥♦ ✜①❡❞ ♣♦✐♥t ✐♥ H✱ ❛♥❞ ❡①❛❝t❧② t✇♦ ✐♥ ∂H✳ ❚❤❡ ❣❡♦❞❡s✐❝ ❥♦✐♥✐♥❣ t❤❡♠ ✭✇❤✐❝❤ ✐s tr❛♥s❧❛t❡❞ ❜② g✮ ✐s ❝❛❧❧❡❞ t❤❡ ❛①✐s ♦❢ g Pr♦♣ ◮ g ✐s ❡❧❧✐♣t✐❝ ✐✛ (tr(g))✷ < ✹ ◮ g ✐s ♣❛r❛❜♦❧✐❝ ✐✛ (tr(g))✷ = ✹ ◮ g ✐s ❤②♣❡r❜♦❧✐❝ ✐✛ (tr(g))✷ > ✹

Pr♦♦❢✳

❙♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ az+b

cz+d = z✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s ✻✹✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈♦♥❥✉❣❛❝② ❝❧❛ss❡s

Pr♦♣ ▲❡t g ∈ PSL✷(R) ◮ g ✐s ❡❧❧✐♣t✐❝ ✐✛ g ✐s ❝♦♥❥✉❣❛t❡❞ ✐♥ PSL✷(R) t♦ cos θ − sin θ sin θ cos θ

• ❢♦r s♦♠❡ θ ∈ R✳

◮ g ✐s ♣❛r❛❜♦❧✐❝ ✐✛ g ✐s ❝♦♥❥✉❣❛t❡❞ ✐♥ PSL✷(R) t♦

◮ z → z + ✶ ✭♠❛tr✐①

• ✶

✶ ✵ ✶

• ✮

◮ ♦r z → z − ✶ ✭♠❛tr✐① ✶ −✶ ✵ ✶

• ✮ ❀

◮ g ✐s ❤②♣❡r❜♦❧✐❝ ✐✛ g ✐s ❝♦♥❥✉❣❛t❡❞ ✐♥ PSL✷(R) t♦ z → λ✷z ✭♠❛tr✐① λ ✵ ✵ ✶/λ

• ✮ ❢♦r s♦♠❡ λ ∈ R>✵✳

✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✺✳✺ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐s♦♠❡tr✐❡s ✻✺✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈❤❛♣t❡r ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

✶✳✶ ❚❤❡ ✉♣♣❡r ❤❛❧❢✲♣❧❛♥❡ ♠♦❞❡❧ ✶✳✷ ❙♦♠❡ ❇❛❝❦❣r♦✉♥❞ ✶✳✸ ❚❤❡ P♦✐♥❝❛ré ❉✐s❦ ♠♦❞❡❧ ✶✳✹ ●❡♦❞❡s✐❝s ✶✳✺ PSL✷(R) ❛♥❞ ✐s♦♠❡tr✐❡s ✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s

✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ✻✻✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❈r♦ss✲r❛t✐♦ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐st❛♥❝❡

Pr♦♣ d(x, y) = ln([p, x, y, q]) ✇❤❡r❡ p ❛♥❞ q ❛r❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ R ♦❢ t❤❡ ❣❡♦❞❡s✐❝ ❧✐♥❡ g t❤r♦✉❣❤ x✱ y✱ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t p✱ x✱ y ❛♥❞ q ♦❝❝✉r ✐♥ t❤✐s ♦r❞❡r ♦♥ g✳

Pr♦♦❢✳

❯s❡ PSL✷(R)✱ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❜♦t❤ s✐❞❡s ❛♥❞ ✐s ✷✲tr❛♥s✐t✐✈❡ ♦♥ R✱ t♦ ♥♦r♠❛❧✐③❡ t♦ p = ✵ ❛♥❞ q = ∞✳ ❊①♦ ✶✳✶✻ ❙❤♦✇ t❤❛t t❤❡ s✐♠✐❧❛r ❢♦r♠✉❧❛ ❤♦❧❞s ✐♥ B

✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ✻✼✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚❤❡ ❤②♣❡r❜♦❧✐❝ r❡✢❡❝t✐♦♥s

◮ ❚❤❡ ✐♥✈❡rs✐♦♥s ✐♥ ❝✐r❝❧❡s ♣❛ss✐♥❣ ❜② i ❛♥❞ ♦rt❤♦❣♦♥❛❧ t♦ R ❛r❡ ❝♦♥❥✉❣❛t❡❞ ❜② Φ : H → B t♦ r❡✢❡❝t✐♦♥s t❤r♦✉❣❤ ❧✐♥❡s t❤r♦✉❣❤ ✵✱ ❤❡♥❝❡ ❛r❡ ❤②♣❡r❜♦❧✐❝ ✐s♦♠❡tr✐❡s ◮ ❈♦♥❥✉❣❛t✐♥❣ ❜② ❛❧❧ s✐♠✐❧❛r✐t✐❡s z → az + b ✇✐t❤ a ∈ R>✵ ❛♥❞ b ∈ R✱ t❤❛t ✐❢ C ✐s ❛ ❣❡♦❞❡s✐❝ ♦❢ H✱ t❤❡♥ t❤❡ ✐♥✈❡rs✐♦♥ ✐♥ t❤❡ ❝✐r❝❧❡ C ✐s ❛♥ ✐s♦♠❡tr② ♦❢ H ✿ ✐t ✐s ❝❛❧❧❡❞ t❤❡ ❤②♣❡r❜♦❧✐❝ r❡✢❡❝t✐♦♥ ✐♥ C✳

✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ✻✽✴✼✶

✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❆♥❣❧❡s ❛♥❞ ❆r❡❛

◮ ✐♥ H ❛♥❞ B✱ ❤②♣❡r❜♦❧✐❝ ❛♥❣❧❡s = ❡✉❝❧✐❞❡❛♥ ❛♥❣❧❡s ✭t❤❡② ❛r❡ ❝♦♥❢♦r♠❛❧ ♠♦❞❡❧s✮ ◮ ❚❤❡ ✭❤②♣❡r❜♦❧✐❝✮ ❛r❡❛ ♦❢ ❛ r❡❣✐♦♥ D ⊂ H ✐s ❞❡✜♥❡❞ ❜② ✐♥t❡❣r❛t✐♥❣ t❤❡ ❛r❡❛ ❡❧❡♠❡♥t (✶/y✷) ❞x ❞y ✭❞❡✜♥❡❞ s✉❝❤ t❤❛t ❛♥② ❤②♣❡r❜♦❧✐❝ ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ♦❢ TzH ❤❛s ✈♦❧ ✶✮ ✿ Area(D) :=

D

✶ y✷ ❞x ❞y ❊①♦ ✶✳✶✼ ❙❤♦✇ t❤❛t ✐s♦♠❡tr✐❡s ♦❢ H ♣r❡s❡r✈❡s t❤❡ ❤②♣❡r❜♦❧✐❝ ❛r❡❛✳

✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ✻✾✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❚r✐❛♥❣❧❡s

◮ ❆ ❤②♣❡r❜♦❧✐❝ tr✐❛♥❣❧❡ ❝♦♥s✐sts ♦❢ t❤r❡❡ ♥♦♥✲❝♦❧❧✐♥❡❛r ♣♦✐♥ts ✐♥ H ∪ ∂H✳ ❆ ✈❡rt❡① ✐♥ ∂H ✐s ❝❛❧❧❡❞ ❛♥ ✐❞❡❛❧ ✈❡rt❡①✳ ◮ ◆❇✿ ■❞❡❛❧ tr✐❛♥❣❧❡s ❛r❡ ❛❧❧ ❝♦♥❣r✉❡♥t

❚❤❡♦r❡♠

✭●❛✉ss✲❇♦♥♥❡t✮ ▲❡t ∆ ❜❡ ❛♥❞ ❤②♣❡r❜♦❧✐❝ tr✐❛♥❣❧❡✱ ❛♥❞ α✱ β✱ γ ❜❡ t❤❡ ❛♥❣❧❡s ♦❢ ∆ ✭t❤❡ ❛♥❣❧❡ ❜❡✐♥❣ ✵ ❢♦r ❛ ✈❡rt❡① ❛t ∞✮✳ ❚❤❡♥ Area(∆) = π − (α + β + γ) ■♥ ♣❛rt✐❝✉❧❛r α + β + γ < π Pr♦♣ ❋♦r ❛❧❧ α, β, γ ≥ ✵ s✉❝❤ t❤❛t α + β + γ < π✱ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❤②♣❡r❜♦❧✐❝ tr✐❛♥❣❧❡ ❛❢ ❛♥❣❧❡s α, β, γ✱ ✉♣ t♦ ✐s♦♠❡tr✐❡s✳

✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ✼✵✴✼✶ ✶ ❚❤❡ ❍②♣❡r❜♦❧✐❝ P❧❛♥❡

❍②♣❡r❜♦❧✐❝ ❝✐r❝❧❡s

Pr♦♣ ■♥ H ❛♥❞ B✱ ❤②♣❡r❜♦❧✐❝ ❝✐r❝❧❡s ❛r❡ ❊✉❝❧✐❞❡❛♥ ❝✐r❝❧❡s ✭❜✉t t❤❡② ❞♦ ♥♦t ❤❛✈❡ t❤❡ s❛♠❡ ❝❡♥t❡r ✐♥ ❣❡♥❡r❛❧✮✳ ✶✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ ❛♥ ❤②♣❡r❜♦❧✐❝ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r ✐s ✷π sinh(r) ✷✳ ❚❤❡ ❛r❡❛ ♦❢ ❛♥ ❤②♣❡r❜♦❧✐❝ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r ✐s ✹π sinh✷( ✶

✷r)

✶✳✻ ❙♦♠❡ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt✐❡s ✼✶✴✼✶