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• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

❆❧❢ ❖♥s❤✉✉s ▼❛r❝❤ ✷✻ ✷✵✶✽

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶ ✴ ✷✻

❈♦♥t❡①t

• ❡♦♠❡tr✐❝ ❋✐❡❧❞s
• ❚❤r♦✉❣❤♦✉t t❤✐s t❛❧❦✱ ✇❡ ✇✐❧❧ ❜❡ ✇♦r❦✐♥❣ ✐♥ ❛ t❤❡♦r② T ✐♥ t❤❡ ❧❛♥❣✉❛❣❡

❝♦♥t❛✐♥✐♥❣ t❤❡ t❤❡♦r② ♦❢ ✜❡❧❞s✳

• ❲❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ♠♦❞❡❧s ♦❢ T ❛r❡ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳ ❚❤✐s ✐s ✜❡❧❞s

✇❤✐❝❤ ❛r❡✿ F ✐s ❛❧❣❡❜r❛✐❝❛❧❧② ❜♦✉♥❞❡❞✳ F ✐s ❞❡✜♥❛❜❧② ❝❧♦s❡❞ ✐♥ ✐ts ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ ✭s♦ ♣❡r❢❡❝t✮✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷ ✴ ✷✻

❈♦♥t❡①t

❆❧❣❡❜r❛✐❝❛❧❧② ❜♦✉♥❞❡❞

❉❡✜♥✐t✐♦♥ ❆ t❤❡♦r② ✐s ❛❧❣❡❜r❛✐❝❛❧❧② ❜♦✉♥❞❡❞ ✐❢✱

• ✐✈❡♥ ❛♥② ❢♦r♠✉❧❛ φ(¯

x, y)✱ t❤❡r❡ ❛r❡ ♣♦❧②♥♦♠✐❛❧s f✶(¯ x, y), . . . , fn(¯ x, y) ∈ Z[¯ x, y] ❙✉❝❤ t❤❛t ✐♥ ❛♥② ♠♦❞❡❧ K ♦❢ T ❛♥❞ ❛♥② ¯ a ❛ t✉♣❧❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ K s✉❝❤ t❤❛t φ(¯ a, K) := {y ∈ K : φ(¯ a, y)} ✐s ✜♥✐t❡✱ t❤❡♥ t❤❡r❡ ✐s ❛♥ ✐♥❞❡① i s✉❝❤ t❤❛t t❤❡ ♣♦❧②♥♦♠✐❛❧ fi(¯ a, y) ✐s ♥♦t ✐❞❡♥t✐❝❛❧❧② ✵ ♦♥ K ❛♥❞ φ(¯ a, K) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ s❡t ♦❢ r♦♦ts ♦❢ fi(¯ a, y) = ✵✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✸ ✴ ✷✻

❈♦♥t❡①t

❊①❛♠♣❧❡s ♦❢ ❣❡♦♠❡tr✐❝ ✜❡❧❞s ✐♥❝❧✉❞❡✿ ❘❡❛❧ ❝❧♦s❡❞ ✜❡❧❞s✳ ♣✲❛❞✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞s✳ Ps❡✉❞♦✜♥✐t❡ ✜❡❧❞s✳ ❇♦✉♥❞❡❞ Ps❡✉❞♦✲❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞s✳ ❇♦✉♥❞❡❞ Ps❡✉❞♦✲r❡❛❧ ❝❧♦s❡❞ ✜❡❧❞s✳ ❇♦✉♥❞❡❞ Ps❡✉❞♦✲♣✲❛❞✐❝❛❧❧② ❝❧♦s❡❞ ✜❡❧❞s✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✹ ✴ ✷✻

❈♦♥t❡①t

❖✉r st❛rt✐♥❣ ♣♦✐♥t

❋❛❝t ❬❍r✉s❤♦✈s❦✐✲P✐❧❧❛②❪ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ T ♦✈❡r ❛ s❡t A✱ ❛♥❞ ❧❡t a, b, c ∈ G(U) ❜❡ ❞✐♠❡♥s✐♦♥ ❣❡♥❡r✐❝ ❡❧❡♠❡♥ts s✉❝❤ t❤❛t a ·G b = c✱ ✇✐t❤ dim(atp(a/A) = dim(b/A) = dim(G) ❛♥❞ s✉❝❤ t❤❛t a ❛♥❞ b ❛r❡ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ♦✈❡r A✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s❡t B ❝♦♥t❛✐♥✐♥❣ A s✉❝❤ t❤❛t a ❛♥❞ b ❛r❡ st✐❧❧ ❛❧❣❡❜r❛✐❝❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ♦✈❡r B✱ ❛ B✲❞❡✜♥❛❜❧❡ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❞✐♠❡♥s✐♦♥✲❣❡♥❡r✐❝ ❡❧❡♠❡♥ts a′, b′, c′ ∈ H(U) s✉❝❤ t❤❛t a′ · b′ = c′ ❛♥❞ acl(Ba) = acl(Ba′)✱ acl(Bb) = acl(Bb′) ❛♥❞ acl(Bc) = acl(Bc′)✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✺ ✴ ✷✻

❈♦♥t❡①t

❍♦✇ ❝❧♦s❡ ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ G ❛♥❞ H❄

✭❍r✉s❤♦✈s❦✐✲P✐❧❧❛②✮

✶ ❲❤❡♥ ♦♥❡ ❤❛s ❛ t♦♣♦❧♦❣②✿ ■❢ F ✐s ❛ r❡❛❧ ❝❧♦s❡❞ ♦r ♣✲❛❞✐❝❛❧❧② ❝❧♦s❡❞

✜❡❧❞✱ ♦♥❡ ❝❛♥ ✉s❡ t❤❡ t♦♣♦❧♦❣② ♦♥ t❤❡ ✜❡❧❞ t♦ ✜♥❞ ❧♦❝❛❧ ✐s♦♠♦r♣❤✐s♠ f ❜❡t✇❡❡♥ ♥❡✐❣❤❜♦r❤♦♦❞s ♦❢ t❤❡ ✐❞❡♥t✐t② a−✶U ❛♥❞ a′−✶U′ ✇❤❡r❡ U ✐s ❛ ✭t✲t♦♣♦❧♦❣②✮ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ a ❛♥❞ U′ ✐s ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ H(F)✳

✷ ■♥ t❤❡ ♣s❡✉❞♦ ✜♥✐t❡ ✜❡❧❞ ❝❛s❡✿ ❯s❡ st❛❜✐❧✐③❡r ♦❢ tp(b, b′/B) ✐♥ t❤❡

❣r♦✉♣ G × H t♦ ✜♥❞ ❛ t②♣❡ ❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣ ♦❢ G × H ✇✐t❤ ❧❛r❣❡ ♣r♦❥❡❝t✐♦♥s ❛♥❞ ✜♥✐t❡ ✜❜❡rs ♦♥ ❜♦t❤ G ❛♥❞ H✳ ❖❜s❡r✈❛t✐♦♥ ■❢ F ✐s R ♦r Qp ❛♥❞ G ✐s ❛ ◆❛s❤ ❣r♦✉♣ t❤❡♥ t❤❡ ❧♦❝❛❧ ✐s♦♠♦r♣❤✐s♠ ✐s ❛ ◆❛s❤ ✐s♦♠♦r♣❤✐s♠✳ ■❢ F = R ♦♥❡ ❝❛♥ t❤❡♥ t❛❦❡ t❤❡ ◆❛s❤ ❝❧♦s✉r❡ ♦❢ U × f (U) ✐♥ G × H ❛♥❞ ❣❡t ❛ ◆❛s❤ ✐s♦❣❡♥② ❜❡t✇❡❡♥ G ❛♥❞ H✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✻ ✴ ✷✻

❈♦♥t❡①t

❇❛rr✐❣❛ ✉s❡❞ t❤❡ ✜rst str❛t❡❣② t♦ ❛❞❛♣t t❤❡ ♣r♦♦❢ ✐♥ t❤❡ r❡❛❧ ❝❛s❡ t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✿ ❚❤❡♦r❡♠ ✭❇❛rr✐❣❛✮ ■❢ G ✐s ❜♦✉♥❞❡❞ ✭✐♥ t❤❡ ♦r❞❡r t♦♣♦❧♦❣②✮ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ ❛ r❡❛❧ ❝❧♦s❡❞ ✜❡❧❞ F t❤❡♥ t❤❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❛ ❧♦❝❛❧ ❤♦♠♦♠♦r♣❤✐s♠ f ❜❡t✇❡❡♥ ❛ ❣❡♥❡r✐❝ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ t❤❡ ✐❞❡♥t✐t② ✐♥ G ❛♥❞ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ H(F)✳ ❙❤❡ t❤❡♥ ✉s❡❞ t❤✐s t♦ ❝❧❛ss✐❢② ❛❧❧ ♦♥❡ ❞✐♠❡♥s✐♦♥❛❧ ❣r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ r❡❛❧ ❝❧♦s❡❞ ✜❡❧❞s ✐♥ t❡r♠s ♦❢ ❞❡✜♥❛❜❧❡ s✉❜s❡ts ♦❢ ✉♥✐✈❡rs❛❧ ❝♦✈❡rs ♦❢ ✭♣♦ss✐❜❧② ✐♥✜♥✐t❡s✐♠❛❧✮ ♥❡✐❣❤❜♦r❤♦♦❞s ♦❢ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣s✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✼ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❲❤✐❧❡ ✇♦r❦✐♥❣ ✐♥ ❛♥❛❧♦❣✉❡s ✐♥ t❤❡ ❝❛s❡ ♦❢ ❜♦✉♥❞❡❞ P❘❈ ✜❡❧❞s ✇❡ ✇❡♥t ❜❛❝❦ t♦ t❤❡ st❛❜✐❧✐③❡r str❛t❡❣②✳ ❉❡✜♥✐t✐♦♥ ✭❙t❛❜✐❧✐③❡r ♦❢ tp(b, b′/B)✱ ✐♥ ❜♦✉♥❞❡❞ P❆❈✮ ▲❡t q(x, x′) ❜❡ ❛ t②♣❡ ✐♥ G × H✱ ❞❡✜♥❡ St(q) : {(x, x′) | (x, x′) · q ∪ q} ✐s ❛ ♥♦♥ ❢♦r❦✐♥❣ ❡①t❡♥s✐♦♥ ♦❢ q✳ Stab(q) ✇✐❧❧ ❜❡ t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② St(q) ■❢ q := tp(b, b′/B) t❤❡♥ Stab(q) = St(q)St(q) ✐s ❛ t②♣❡ ❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣ ♦❢ G × H✱ ✇❤✐❝❤ ♠❛♣s ✇✐t❤ ✜♥✐t❡ ✜❜❡rs ♦♥t♦ t②♣❡ ❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣s ♦❢ ❜♦✉♥❞❡❞ ✐♥❞❡① ♦❢ G ❛♥❞ H✱ r❡s♣❡❝t✐✈❡❧②✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✽ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❖♥❡ ♥❡❡❞s t❤❡ ❙✶ ♣r♦♣❡rt② ♦♥ t❤❡ ❢♦r❦✐♥❣ ✐❞❡❛❧✿ ❉❡✜♥✐t✐♦♥ ▲❡t µ ❜❡ ❛♥ A✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ ♦♥ ❞❡✜♥❛❜❧❡ s❡ts✳ ❚❤❡♥ µ ❤❛s t❤❡ ❙✶ ♣r♦♣❡rt② ✐❢ ❣✐✈❡♥ ❛♥② φ(x, a✵) ❛♥❞ φ(x, a✶)✱ ❢♦r a✵ ❛♥❞ a✶ st❛rt✐♥❣ ❛♥ A✲✐♥❞✐s❝❡r♥✐❜❧❡ s❡q✉❡♥❝❡✱ φ(x, a✵) ∧ φ(x, a✶) ∈ µ ⇒ φ(x, a✵) ∈ µ.

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✾ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❲❡ ✇✐❧❧ ✜① ❛ ♠♦❞❡❧ M ❛♥❞ ❧❡t G ❜❡ ❛♥ M✲❞❡✜♥❛❜❧❡ ❣r♦✉♣✳ ❲❡ ✇✐❧❧ ❢r♦♠ ♥♦✇ ♦♥ ✇♦r❦ ✇✐t❤ ❛♥ ✐❞❡❛❧ µ ♦❢ ❞❡✜♥❛❜❧❡ s✉❜s❡ts ♦❢ G ✇❤✐❝❤ ✐s M✲✐♥✈❛r✐❛♥t ❛♥❞ ✇❤✐❝❤ ✐s ✐♥✈❛r✐❛♥t ❜② ❧❡❢t ✭❛♥❞ s♦♠❡t✐♠❡s ❧❡❢t ❛♥❞ r✐❣❤t✮ tr❛♥s❧❛t✐♦♥s ❜② ❡❧❡♠❡♥ts ♦❢ G✳ ❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❛ t②♣❡ p(x) ✐♥ G ✐s µ✲✇✐❞❡ ✭♦r ❥✉st ✏✇✐❞❡✑✮ ✐❢ ✐t ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛ s❡t D ∈ µ✳ ❉❡✜♥✐t✐♦♥ ■❢ q ❛♥❞ r ❛r❡ ✇✐❞❡ t②♣❡s✱ t❤❡♥ ✇❡ ❞❡✜♥❡✿ St(q, r) := {g : gp ∪ r ✐s ✇✐❞❡}✳ St(p) = St(p, p) Stab(p) t❤❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② St(p)✳

❏✉♠♣ t♦ t❤❡♦r❡♠ ❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✵ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❋❛❝t ✭❍r✉s❤♦✈s❦✐✮ ▲❡t µ ❜❡ ❛♥ M✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ ♦♥ G st❛❜❧❡ ✉♥❞❡r ❧❡❢t ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ▲❡t X ⊆ G ❜❡ ❛♥ M✲❞❡✜♥❛❜❧❡ s❡t s✉❝❤ t❤❛t µ ✐s ❙✶ ♦♥ X(X)−✶X✳ ▲❡t q ❜❡ ❛ ✇✐❞❡ t②♣❡ ♦✈❡r M ❝♦♥❝❡♥tr❛t✐♥❣ ♦♥ X✳ ❆ss✉♠❡ ✭❋✮ ❚❤❡r❡ ❛r❡ a, b | = q s✉❝❤ t❤❛t tp(a/Mb) ❛♥❞ tp(b/Ma) ❛r❡ ❜♦t❤ ♥♦♥✲❢♦r❦✐♥❣ ♦✈❡r M✳ ❚❤❡♥ t❤❡r❡ Stab(q) ✐s ❛ ✇✐❞❡ t②♣❡✲❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣ ♦❢ G ❛♥❞ Stab(q) = (q−✶q)✷

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✶ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❆❧❧ ♦✉r ❡①❛♠♣❧❡s ❛r❡ ◆❚P✷✱ ✇❤❡r❡ ❈♦♥❞✐t✐♦♥ ✭❋✮ ✐s ❛❧✇❛②s s❛t✐s✜❡❞✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✷ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❲❤❛t ✇❡ ✇❛♥t t♦ ♣r♦✈❡

❚❤❡♦r❡♠ ❬◆❚P✷ ❱❡rs✐♦♥❪ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ ❛ ω✲s❛t✉r❛t❡❞ ♠♦❞❡❧ M ♦❢ ❛♥ ◆❚P✷ t❤❡♦r② T ✇❤✐❝❤ ✐s ❛ ❣❡♦♠❡tr✐❝ ✜❡❧❞✳ ❆ss✉♠❡ t❤❛t T ❛❞♠✐ts ❛♥ M✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ µG ♦♥ G✱ st❛❜❧❡ ✉♥❞❡r ❧❡❢t ❛♥❞ r✐❣❤t ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ s✉❝❤ t❤❛t µG ✐s ❙✶ ♦♥ G✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❛ ❞❡✜♥❛❜❧❡ ✜♥✐t❡✲t♦✲♦♥❡ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❛ t②♣❡✲❞❡✜♥❛❜❧❡ ✇✐❞❡ s✉❜❣r♦✉♣ D ♦❢ G t♦ H(M)✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ G ✐s ❛♠❡♥❛❜❧❡ t❤❡♥ D ✇✐❧❧ ❤❛✈❡ ❜♦✉♥❞❡❞ ✐♥❞❡① ❛♥❞ ✇❡ ❝❛♥ ❣❡t ❛ ❧♦❝❛❧ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❛ ❣❡♥❡r✐❝ s✉❜s❡t ♦❢ G ✐♥t♦ H✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✸ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❆tt❡♠♣t t♦ ✉s❡ ❍r✉s❤♦✈s❦✐✬s t❤❡♦r❡♠

• ❲❡ ❤❛✈❡ a, a′✱ b, b′ ❛♥❞ c, c′ ❛s ✐♥ ❛♥② ❣❡♦♠❡tr✐❝ ✜❡❧❞✳ ❆ss✉♠❡ G ✐s

❞❡✜♥❛❜❧❡ ✐♥ ❛ ◆❚P✷ t❤❡♦r② ✇✐t❤ µG ❛s ✐♥ t❤❡ t❤❡♦r❡♠✳ ✭❚❤❡ ③❡r♦ ♠❡❛s✉r❡ ✐❞❡❛❧ ✐♥ ❛♥ ❛♠❡♥❛❜❧❡ ❣r♦✉♣ ✐s ❛❧✇❛②s S✶ ❛♥❞ µG ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ❜✐✲✐♥✈❛r✐❛♥t✳✮

• ❲❡ ♠❛② ❛ss✉♠❡ tp(a/M), tp(b/M) ❛♥❞ tp(a/Mb) ❛r❡ ❛❧❧ µ✲✇✐❞❡ t②♣❡

p(x)✳

• t❤❡ t②♣❡ p = tp(a, a′/M) ❤❛s ✜♥✐t❡ ✜❜❡rs✳
• ❲❡ ❝❛♥ tr② t♦ ♣✉❧❧ ❜❛❝❦ t❤❡ ③❡r♦ ✐❞❡❛❧ ♦❢ µG ✭s❡ts t❤❛t ♣r♦❥❡❝t ✐♥ G t♦ ❛

s❡t ✐♥ µG✮✳

• µ ✇✐❧❧ ❜❡ ❙✶ ♦♥ ❞❡✜♥❛❜❧❡ ❛♥② ❞❡✜♥❛❜❧❡ s✉❜s❡t X ♦❢ G × H t❤❛t ❤❛✈❡

✜♥✐t❡ ✜❜❡rs ✐♥ H✳ ❈❛♥ ✇❡ ✜♥❞ ❛ s❡t X✵ ❝♦♥t❛✐♥✐♥❣ tp(a, a′) s✉❝❤ t❤❛t µ ❙✶ ♦♥ X✵(X✵)−✶❄ ❉♦❡s pp−✶ ❤❛✈❡ ✜♥✐t❡ ✜❜❡rs ✭s❛②✱ ♦✈❡r t❤❡ ✐❞❡♥t✐t② eG✮❄

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✹ ✴ ✷✻

❚❤❡ ❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠

❙t❛❜✐❧✐③❡r ❚❤❡♦r❡♠ ✇✐t❤ λ

• ❲❡ ❦❡❡♣ µ ❜❡ s❡ts t❤❛t ♣r♦❥❡❝t ♦♥t♦ µG✲✇✐❞❡ s❡ts ✐♥ G✳
• ❲❡ ❝❤♦♦s❡ ❡①♣❧✐❝✐t❧② ❛ s❡❝♦♥❞ ✐❞❡❛❧ λ ♦❢ ❞❡✜♥❛❜❧❡ s❡ts ❝♦♥t❛✐♥✐♥❣ tp(a, a′)

✐♥ ✇❤✐❝❤ µ ✐s ✭❙✶✮✳ ❙❡ts t❤❛t ♣r♦❥❡❝t ✇✐t❤ ✜♥✐t❡ ✜❜❡rs ♦♥ ❜♦t❤ G ❛♥❞ H✳ ❚❤❡♦r❡♠ ❬◆❚P✷ ❱❡rs✐♦♥❪ ▲❡t µ ❛♥❞ λ ❜❡ M✲✐♥✈❛r✐❛♥t ✐❞❡❛❧s ♦♥ G✱ ✐♥✈❛r✐❛♥t ✉♥❞❡r ❧❡❢t ❛♥❞ r✐❣❤t ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ s✉❝❤ t❤❛t µ ✐s ❙✶ ✐♥ ❛♥② X ∈ λ✳ ❆ss✉♠❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ✇✐❞❡ ✭◆❖❚ ✐♥ µ✮ ❛♥❞ ♠❡❞✐✉♠ ✭■◆ λ✮t②♣❡ p ✐♥ G ❛♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ s❛t✐s✜❡❞✿ ✭❆✮ ❢♦r ❛♥② t②♣❡s q✱ r✱ ✐❢ ❢♦r s♦♠❡ (c, d) | = q ×nf r✱ tp(cd/M) ♦r tp(dc/M) ✐s ♠❡❞✐✉♠✱ t❤❡♥ q ✐s ♠❡❞✐✉♠❀ ✭❇✮ ❢♦r ❛♥② (a, b) ∈ p ×nf p✱ tp(a−✶b/M) ✐s ♠❡❞✐✉♠❀ ❚❤❡♥ Stab(p) = St(p)✷ = (pp−✶)✷ ✐s ❛ ❝♦♥♥❡❝t❡❞ t②♣❡✲❞❡✜♥❛❜❧❡✱ ✇✐❞❡ ❛♥❞ ♠❡❞✐✉♠ ❣r♦✉♣✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✺ ✴ ✷✻

❆♣♣❧②✐♥❣ ✐t t♦ ❣r♦✉♣s ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s

❚❤❡♦r❡♠ ❬◆❚P✷ ❱❡rs✐♦♥❪ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ ❛ ω✲s❛t✉r❛t❡❞ ♠♦❞❡❧ M ♦❢ ❛♥ ◆❚P✷ t❤❡♦r② T✳ ❆ss✉♠❡ t❤❛t T ❛❞♠✐ts ❛♥ M✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ µG ♦♥ G✱ st❛❜❧❡ ✉♥❞❡r ❧❡❢t ❛♥❞ r✐❣❤t ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ s✉❝❤ t❤❛t µG ✐s ❙✶ ✐♥ G✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❛ ❞❡✜♥❛❜❧❡ ✜♥✐t❡✲t♦✲♦♥❡ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❛ t②♣❡✲❞❡✜♥❛❜❧❡ ✇✐❞❡ s✉❜❣r♦✉♣ D ♦❢ G t♦ H(M)✳

❏✉♠♣ t♦ ❝♦r♦❧❧❛r✐❡s ❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✻ ✴ ✷✻

❆♣♣❧②✐♥❣ ✐t t♦ ❣r♦✉♣s ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s

• ❚❤❡ ✐❞❡❛❧ µ ♦♥ G × H ✐❢ ❛♥❞ ♦♥❧② ✐❢ π✶(D) ∈ µG✳

µ ✐s M✲✐♥✈❛r✐❛♥t ❛♥❞ ✐♥✈❛r✐❛♥t ✉♥❞❡r ❧❡❢t ❛♥❞ r✐❣❤t tr❛♥s❧❛t✐♦♥s✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ µ✲✇✐❞❡ ❛s ✏✇✐❞❡✑✳

• ❚❤❡ ✐❞❡❛❧ λ ✐s t❤❡ s❡t ♦❢ s✉❜s❡ts X ♦❢ G × H ❢♦r ✇❤✐❝❤ t❤❡ ♣r♦❥❡❝t✐♦♥s t♦

G ❛♥❞ H ❡❛❝❤ ❤❛✈❡ ✜♥✐t❡ ✜❜❡rs✳ ❆ s❡t ✐♥ λ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ λ✲♠❡❞✐✉♠✳

• p = tp(a, a′/M)✳

p ✐s ✇✐❞❡ ❛♥❞ λ✲♠❡❞✐✉♠ ✭❆✮ ❛♥❞ ✭❇✮ ❛❧s♦ ❤♦❧❞✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✼ ✴ ✷✻

❆♣♣❧②✐♥❣ ✐t t♦ ❣r♦✉♣s ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s

❈❧❛✐♠ ❈♦♥❞✐t✐♦♥ ✭❆✮ ❤♦❧❞s✿ ■❢ p, q ❛r❡ t✇♦ t②♣❡s ✐♥ G × H ❛♥❞ ✇❡ ❤❛✈❡ (g, h) | = p ×nf q s✉❝❤ t❤❛t ❡✐t❤❡r tp(gh/M) ♦r tp(hg/M) ✐s λ✲♠❡❞✐✉♠✱ t❤❡♥ p ✐s λ✲♠❡❞✐✉♠✳ Pr♦♦❢✳ ▲❡t (g✵, g✶) ❛♥❞ (h✵, h✶) ❜❡ s✉❝❤ t❤❛t tp((h✵, h✶)/M(g✵, g✶)) ❞♦❡s ♥♦t ❢♦r❦ ♦✈❡r M✳ ❙✐♥❝❡ g✵h✵ ∈ acl(Mg✶h✶) ✇❡ ❤❛✈❡ g✵ ∈ acl(Mg✶h✵h✶)✳ ❆s tp(h✵h✶/Mg✵g✶) ❞♦❡s ♥♦t ❢♦r❦ ♦✈❡r M✱ t❤✐s ✐♠♣❧✐❡s t❤❛t g✵ ∈ acl(Mg✶)✳ ■♥ t❤❡ s❛♠❡ ✇❛② ✇❡ ❣❡t g✶ ∈ acl(Mg✵)✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✽ ✴ ✷✻

❆♣♣❧②✐♥❣ ✐t t♦ ❣r♦✉♣s ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s

❙♦ Stab(p) ✐s ❛ t②♣❡ ❞❡✜♥❛❜❧❡ s✉❜❣r♦✉♣ K ♦❢ G × H ✇✐t❤ ✜♥✐t❡ ✜❜❡rs ♦♥ ❜♦t❤ ♣r♦❥❡❝t✐♦♥s ❛♥❞ s✉❝❤ t❤❛t t❤❡ ♣r♦❥❡❝t✐♦♥ t♦ G ❤❛s ❜♦✉♥❞❡❞ ✐♥❞❡①✳ ❆s K fin = π−✶

✶ (eG) ∩ K✳ ❚❤❡♥ K✶ ✐s ✜♥✐t❡ ❛♥❞ ♥♦r♠❛❧ ✐♥ K✳ ❆s K ✐s

❝♦♥♥❡❝t❡❞✱ K✶ ✐s ❝❡♥tr❛❧ ✐♥ K✳ ▲❡t C ≤ H ❜❡ t❤❡ ❝❡♥tr❛❧✐③❡r ♦❢ π✷(K fin) ✐♥ H✱ ✇❤✐❝❤ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ s✉❜❣r♦✉♣ ♦❢ H ♦❢ ✜♥✐t❡ ✐♥❞❡①✳ ❲❡ r❡♣❧❛❝❡ H ❜② C/π✷(K fin) ✇❤✐❝❤ ✐s ❛❣❛✐♥ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ ✭❞❡✜♥❡❞ ♦✈❡r t❤❡ s❛♠❡ ♣❛r❛♠❡t❡rs ❛s H ❛♥❞ K fin✮✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✶✾ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❚❤❡♦r❡♠ ❬◆❚P✷ ❱❡rs✐♦♥❪ ▲❡t G ❜❡ ❛ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ ❛ ω✲s❛t✉r❛t❡❞ ♠♦❞❡❧ M ♦❢ ❛♥ ◆❚P✷ t❤❡♦r② T✳ ❆ss✉♠❡ t❤❛t T ❛❞♠✐ts ❛♥ M✲✐♥✈❛r✐❛♥t ✐❞❡❛❧ µG ♦♥ G✱ st❛❜❧❡ ✉♥❞❡r ❧❡❢t ❛♥❞ r✐❣❤t ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❛♥❞ s✉❝❤ t❤❛t µG ✐s ❙✶ ✐♥ G✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❛ ❞❡✜♥❛❜❧❡ ✜♥✐t❡✲t♦✲♦♥❡ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❛ t②♣❡✲❞❡✜♥❛❜❧❡ ✇✐❞❡ s✉❜❣r♦✉♣ D ♦❢ G t♦ H(M)✳ ❈♦r♦❧❧❛r② ▲❡t G ❜❡ ❛ t♦rs✐♦♥ ❢r❡❡ ❣r♦✉♣ ❞❡✜♥❛❜❧❡ ✐♥ ❛ r❡❛❧ ❝❧♦s❡❞ ✜❡❧❞ R✳ ❚❤❡♥ t❤❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H s✉❝❤ t❤❛t G ✐s ❞❡✜♥❛❜❧② ✐s♦♠♦r♣❤✐❝ t♦ ❛ s✉❜❣r♦✉♣ ♦❢ H(R) ♦❢ ✜♥✐t❡ ✐♥❞❡①✳ ❇② s♦❧✈❛❜❧❡ G ✐s ❛♠❡♥❛❜❧❡✱ ❛♥❞ ❜② t♦rs✐♦♥ ❢r❡❡ G = G ✵✵✳ ❙♦ D = G✳ ❇② t♦rs✐♦♥ ❢r❡❡✱ t❤❡ ❦❡r♥❡❧ ♦❢ t❤❡ ♠❛♣ ✐s t❤❡ ✐❞❡♥t✐t②✳ ❚❤✐s ❣✐✈❡s ❛♥ ✐♥❥❡❝t✐♦♥ ❢r♦♠ G t♦ H(R)✳ ❇✉t ❛♥② ❞✐♠❡♥s✐♦♥ n s✉❜❣r♦✉♣ ♦❢ R ❤❛s ✜♥✐t❡ ✐♥❞❡①✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✵ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❋♦r G ❞❡✜♥❛❜❧② ❛♠❡♥❛❜❧❡ ❞❡✜♥❛❜❧❡ ✐♥ Qp ✇❡ ❣❡t ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ H ❛♥❞ ❛ ❞❡✜♥❛❜❧❡ ✜♥✐t❡✲t♦✲♦♥❡ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❛ t②♣❡✲❞❡✜♥❛❜❧❡ ✇✐❞❡ s✉❜❣r♦✉♣ D ♦❢ G t♦ H(M)✳ ❆s ✐♥ t❤❡ t❤❡♦r❡♠✱ ❜② ❣♦✐♥❣ t♦ ❛ ✜♥✐t❡ ✐♥❞❡① s✉❜❣r♦✉♣ ♦❢ G ❛♥❞ ♠♦❞❞✐♥❣ ❜♦✉t ❜② ❛ ✜♥✐t❡ s✉❜❣r♦✉♣ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ✐s ♦♥❡ t♦ ♦♥❡✳ ❚❤✐s ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❤❛✈❡ ❝♦♥t✐♥✉♦✉s ✐♠❛❣❡ ✭❜❡❝❛✉s❡ ❛♥② ❞❡✜♥❛❜❧❡ s❡t ❤❛s ❛♥ ♦♣❡♥ s✉❜s❡t ♦❢ ❧❛r❣❡ ❝♦❞✐♠❡♥s✐♦♥✮✳ ❚❤❡♦r❡♠ ✶ ❬✭T | = PRC✱▼♦♥t❡♥❡❣r♦✲❖✳✲❙✐♠♦♥✮ ✭T | = Qp✱ ✇✐t❤ ❆❝♦st❛✮❪ ▲❡t G ❜❡ ❞❡✜♥❛❜❧② ❛♠❡♥❛❜❧❡✳ ❚❤❡♥ t❤❡r❡ ✐s K ≤ G✶ ≤ G s✉❝❤ t❤❛t G✶ ❤❛s ✜♥✐t❡ ✐♥❞❡① ✐♥ G✱ K ✐s ✜♥✐t❡ ❛♥❞ ❝❡♥tr❛❧ ✐♥ G✶ ❛♥❞ s✉❝❤ t❤❛t G✶/K ✐s ❛ ❞❡✜♥❛❜❧❡ ▲✐❡ ❣r♦✉♣ ✇✐t❤ ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ♦❢ ♦♣❡♥ s✉❜s❡ts ❞✐✛❡♦♠♦r♣❤✐❝ ✇✐t❤ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✶ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❚❤❡♦r❡♠ ✭✇✐t❤ ❆❝♦st❛✮ ❆♥② ❛♠❡♥❛❜❧❡ ❣r♦✉♣ G ❞❡✜♥❛❜❧❡ ✐♥ Qp✳ ❚❤❡♥ t❤❡r❡ ✐s K ≤ G✶ ≤ G s✉❝❤ t❤❛t G✶ ❤❛s ✜♥✐t❡ ✐♥❞❡① ✐♥ G✱ K ✐s ✜♥✐t❡ ❛♥❞ ❝❡♥tr❛❧ ✐♥ G✶ ❛♥❞ s✉❝❤ t❤❛t G✶/K ✐s ❛ ♣✲❛❞✐❝ ▲✐❡ ❣r♦✉♣ ✇✐t❤ ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ♦❢ ♦♣❡♥ s✉❜s❡ts ❞✐✛❡♦♠♦r♣❤✐❝ ✇✐t❤ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✷ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❚❤❡♦r❡♠ ✭✇✐t❤ ❆❝♦st❛✮ ❆♥② ❛♠❡♥❛❜❧❡ ❣r♦✉♣ G ❞❡✜♥❛❜❧❡ ✐♥ Qp✳ ❚❤❡♥ G ✐s ❛ ♣✲❛❞✐❝ ▲✐❡ ❣r♦✉♣ ✇✐t❤ ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ♦❢ ♦♣❡♥ s✉❜s❡ts ❞✐✛❡♦♠♦r♣❤✐❝ ✇✐t❤ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣✳ ❈♦r♦❧❧❛r② ✭❆♣♣❧②✐♥❣ r❡s✉❧ts r❡♣♦rt❡❞ ❜② P✐❧❧❛②✲❨❛♦✮ ■❢ G ✐s ❛ ❞❡✜♥❛❜❧❡ ❣r♦✉♣ ♦❢ ❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ t❤❡ ♣✲❛❞✐❝s✱ t❤❡♥ ✐t ❛❞♠✐ts ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ❜② s✉❜s❡ts ❛❧❧ ❞✐✛❡♦♠♦r♣❤✐❝ t♦ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ ♦❢ ❞✐♠❡♥s✐♦♥ ♦♥❡✳

• ❖♥❣♦✐♥❣ ✇♦r❦ ✭❆❝♦st❛✮✳ ❖♥❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡✱ ✉s✐♥❣ ❖✳✲P✐❧❧❛② ❛♥❞ r❡s✉❧ts

❢r♦♠ ❱♦❥❞❛♥✐ s❤♦✇ t❤❡ ✏▼❛✐♥ ❈♦♥❥❡❝t✉r❡✑ ❢r♦♠ ❖✳✲P✐❧❧❛② ❢♦r ✶✲❞✐♠❡♥s✐♦♥❛❧ ❣r♦✉♣s✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✷ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❚❤❡♦r❡♠ ✭✇✐t❤ ❆❝♦st❛✮ ❆♥② ❛♠❡♥❛❜❧❡ ❣r♦✉♣ G ❞❡✜♥❛❜❧❡ ✐♥ Qp✳ ❚❤❡♥ G ✐s ❛ ♣✲❛❞✐❝ ▲✐❡ ❣r♦✉♣ ✇✐t❤ ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ♦❢ ♦♣❡♥ s✉❜s❡ts ❞✐✛❡♦♠♦r♣❤✐❝ ✇✐t❤ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣✳ ❈♦r♦❧❧❛r② ✭❆♣♣❧②✐♥❣ r❡s✉❧ts r❡♣♦rt❡❞ ❜② P✐❧❧❛②✲❨❛♦✮ ■❢ G ✐s ❛ ❞❡✜♥❛❜❧❡ ❣r♦✉♣ ♦❢ ❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ t❤❡ ♣✲❛❞✐❝s✱ t❤❡♥ ✐t ❛❞♠✐ts ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ❜② s✉❜s❡ts ❛❧❧ ❞✐✛❡♦♠♦r♣❤✐❝ t♦ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ ♦❢ ❞✐♠❡♥s✐♦♥ ♦♥❡✳

• ❖♥❣♦✐♥❣ ✇♦r❦ ✭❆❝♦st❛✮✳ ❖♥❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡✱ ✉s✐♥❣ ❖✳✲P✐❧❧❛② ❛♥❞ r❡s✉❧ts

❢r♦♠ ❱♦❥❞❛♥✐ s❤♦✇ t❤❡ ✏▼❛✐♥ ❈♦♥❥❡❝t✉r❡✑ ❢r♦♠ ❖✳✲P✐❧❧❛② ❢♦r ✶✲❞✐♠❡♥s✐♦♥❛❧ ❣r♦✉♣s✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✷ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❚❤❡♦r❡♠ ✭✇✐t❤ ❆❝♦st❛✮ ❆♥② ❛♠❡♥❛❜❧❡ ❣r♦✉♣ G ❞❡✜♥❛❜❧❡ ✐♥ Qp✳ ❚❤❡♥ G ✐s ❛ ♣✲❛❞✐❝ ▲✐❡ ❣r♦✉♣ ✇✐t❤ ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ♦❢ ♦♣❡♥ s✉❜s❡ts ❞✐✛❡♦♠♦r♣❤✐❝ ✇✐t❤ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣✳ ❈♦r♦❧❧❛r② ✭❆♣♣❧②✐♥❣ r❡s✉❧ts r❡♣♦rt❡❞ ❜② P✐❧❧❛②✲❨❛♦✮ ■❢ G ✐s ❛ ❞❡✜♥❛❜❧❡ ❣r♦✉♣ ♦❢ ❞✐♠❡♥s✐♦♥ ♦♥❡ ✐♥ t❤❡ ♣✲❛❞✐❝s✱ t❤❡♥ ✐t ❛❞♠✐ts ❛ ✜♥✐t❡ ❝♦✈❡r✐♥❣ ❜② s✉❜s❡ts ❛❧❧ ❞✐✛❡♦♠♦r♣❤✐❝ t♦ ❛♥ ♦♣❡♥ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ❛♥ ❛❧❣❡❜r❛✐❝ ❣r♦✉♣ ♦❢ ❞✐♠❡♥s✐♦♥ ♦♥❡✳

• ❖♥❣♦✐♥❣ ✇♦r❦ ✭❆❝♦st❛✮✳ ❖♥❡ s❤♦✉❧❞ ❜❡ ❛❜❧❡✱ ✉s✐♥❣ ❖✳✲P✐❧❧❛② ❛♥❞ r❡s✉❧ts

❢r♦♠ ❱♦❥❞❛♥✐ s❤♦✇ t❤❡ ✏▼❛✐♥ ❈♦♥❥❡❝t✉r❡✑ ❢r♦♠ ❖✳✲P✐❧❧❛② ❢♦r ✶✲❞✐♠❡♥s✐♦♥❛❧ ❣r♦✉♣s✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✷ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

▼♦r❡ ❖♥❣♦✐♥❣ ✇♦r❦ ✶✿ PP❈ ✜❡❧❞s

• ✭✇✐t❤ ▼♦♥t❡♥❡❣r♦ ❛♥❞ ❙✐♠♦♥✮ ❚❤❡♦r❡♠ ✶ s❤♦✉❧❞ ❤♦❧❞ ❢♦r ❜♦✉♥❞❡❞ PP❈

✜❡❧❞s✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✸ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

▼♦r❡ ❖♥❣♦✐♥❣ ✇♦r❦ ✷✿ ❙✐♠♣❧❡ ❣r♦✉♣s ✭❆❢t❡r P❡t❡r③✐❧✲P✐❧❧❛②✲❙t❛r❝❤❡♥❦♦✮✳

❆♠❡♥❛❜❧❡ ❣r♦✉♣s t❡♥❞ t♦ ❜❡ ❝♦♠♣❧❡♠❡♥t❡❞ ❜② s✐♠♣❧❡ ❣r♦✉♣s✳ ■♥ s♦♠❡ P❘❈ ✜❡❧❞s ✭❧✐❦❡ t❤♦s❡ st✉❞✐❡❞ ❜② ✈❛♥ ❞❡♥ ❉r✐❡s✮ ♦♥❡ ❤❛s ❛ t✲t♦♣♦❧♦❣②✳ ❖♥❡ ❝❛♥ r❡❝♦✈❡r t❤❡ ❧♦❝❛❧ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❍r✲P✐✳

• ❖♥❡ ❤❛s ❛ ❧♦❝❛❧ ❣r♦✉♣ ❤♦♠♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ G ❛♥❞ H✳ ❖♥❡ ❧♦♦❦s ❛t

t❤❡ ❛❝t✐♦♥s t❤✐s ✐♠♣❧② ✐♥ t❤❡ t❛♥❣❡♥t s♣❛❝❡ T H

e

✭t❤❡ ❛❞❥♦✐♥t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ H ❛♥❞ t❤❡ ❛❝t✐♦♥ ❜② ❝♦♥❥✉❣❛t✐♦♥ ♦❢ G ✇❤✐❝❤ ❝♦♠❡s ❢r♦♠ t❤❡ ❧♦❝❛❧ ❤♦♠♦♠♦r♣❤✐s♠✮✳ ❇② s✐♠♣❧✐❝✐t② t❤❡s❡ ❛r❡ ❢❛✐t❤❢✉❧✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✹ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❋✉t✉r❡ ✇♦r❦ ✸✿ ❈♦♠❜✐♥✐♥❣

❍♦✇ ♠✉❝❤ ❝❛♥ ♦♥❡ ❝♦♠❜✐♥❡ t❤❡ s✐♠♣❧❡ ❛♥❞ ❛♠❡♥❛❜❧❡ ❝❛s❡s t♦ ❣❡t ❛ ♠♦r❡ ❜r♦❛❞ t❤❡♦r❡♠✳

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✺ ✴ ✷✻

❆♣♣❧✐❝❛t✐♦♥s

❚❤❛♥❦s✦

❆❧❢ ❖♥s❤✉✉s

• r♦✉♣s ❞❡✜♥❛❜❧❡ ✐♥ ❣❡♦♠❡tr✐❝ ✜❡❧❞s✳

▼❛r❝❤ ✷✻ ✷✵✶✽ ✷✻ ✴ ✷✻