Growth rates of braid monoids with many generators Ramón Flores 1 , Juan González-Meneses 1 & Vincent Jugé 2 1: Universidad de Sevilla – 2: Université Paris-Est Marne-la-Vallée (LIGM) 20/06/2019 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Contents Growth rates of trace monoids 1 Growth rates of trace monoids: a first proof 2 Growth rates of braid monoids: a first proof 3 Growth rates of trace and braid monoids: an algebraic proof 4 Conclusion 5 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Heaps of pieces vs Trace monoids Heap of pieces S 1 S 3 S 2 S 1 S 4 1 2 3 4 5 S 3 S 1 S 3 S 1 S 3 S 4 S 2 S 2 S 1 S 4 S 1 S 2 S 1 S 4 S 1 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Heaps of pieces vs Trace monoids Heap of pieces Trace monoid S 1 S 3 F ` B ˇ S i S j “ S j S i ˇ S 2 T 4 “ S 1 , S 2 , S 3 , S 4 ˇ if i ‰ j ˘ 1 ˇ S 1 S 4 1 2 3 4 5 S 3 S 1 S 3 S 1 S 3 S 4 S 2 S 2 S 1 S 4 S 1 S 2 S 1 S 4 S 1 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 S 1 ¨ S 4 ¨ S 2 ¨ S 1 ¨ S 3 S 4 ¨ S 1 ¨ S 2 ¨ S 3 ¨ S 1 S 1 ¨ S 2 ¨ S 1 ¨ S 4 ¨ S 3 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Heaps of pieces vs Trace monoids Heap of pieces Trace monoid S 1 S 3 F ` B ˇ S i S j “ S j S i ˇ S 2 T 4 “ S 1 , S 2 , S 3 , S 4 ˇ if i ‰ j ˘ 1 ˇ S 1 S 4 1 2 3 4 5 | h | “ #pieces in h | τ | “ #generators needed to write τ Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Heaps of pieces vs Trace monoids Heap of pieces Trace monoid S 1 S 3 F ` B ˇ S i S j “ S j S i ˇ S 2 T 4 “ S 1 , S 2 , S 3 , S 4 ˇ if i ‰ j ˘ 1 ˇ S 1 S 4 1 2 3 4 5 | h | “ #pieces in h | τ | “ #generators needed to write τ λ 4 , k “ #heaps of size k λ 4 , k “ # traces of size k Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Heaps of pieces vs Trace monoids Heap of pieces Trace monoid S 1 S 3 F ` B ˇ S i S j “ S j S i ˇ S 2 T 4 “ S 1 , S 2 , S 3 , S 4 ˇ if i ‰ j ˘ 1 ˇ S 1 S 4 1 2 3 4 5 | h | “ #pieces in h | τ | “ #generators needed to write τ λ 4 , k “ #heaps of size k λ 4 , k “ # traces of size k How does λ 4 , k behave when k Ñ `8 ? Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of a finitely generated monoid In a monoid M generated by a finite family F , | τ | “ #generators (in F ) needed to write τ m k “ #elements of size k Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of a finitely generated monoid In a monoid M generated by a finite family F , | τ | “ #generators (in F ) needed to write τ m k “ #elements of size k Lemma: m k ` ℓ ď m k m ℓ ( log m k is sub-additive ) Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of a finitely generated monoid In a monoid M generated by a finite family F , | τ | “ #generators (in F ) needed to write τ m k “ #elements of size k Lemma: m k ` ℓ ď m k m ℓ ( log m k is sub-additive ) Corollary: The sequence m 1 { k converges! k Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of a finitely generated monoid In a monoid M generated by a finite family F , | τ | “ #generators (in F ) needed to write τ m k “ #elements of size k Lemma: m k ` ℓ ď m k m ℓ ( log m k is sub-additive ) Corollary: The sequence m 1 { k converges! k Proof: If m k “ 0 for some k ě 0, then m ℓ “ 0 for all ℓ ě k . Otherwise, set x k “ p log m k q{ k ě 0 and X k “ max t x 1 , . . . , x k u . For all ℓ ď k and q ě 1, we have q Ñ8 x qk ` ℓ ď p k q x k ` ℓ x ℓ q{p q k ` ℓ q ď x k ` X k { q Ý Ý Ý Ñ x k and thus lim sup k Ñ8 x k ď x k . Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of a finitely generated monoid In a monoid M generated by a finite family F , | τ | “ #generators (in F ) needed to write τ m k “ #elements of size k Lemma: m k ` ℓ ď m k m ℓ ( log m k is sub-additive ) Corollary: The sequence m 1 { k converges! k Proof: If m k “ 0 for some k ě 0, then m ℓ “ 0 for all ℓ ě k . Otherwise, set x k “ p log m k q{ k ě 0 and X k “ max t x 1 , . . . , x k u . For all ℓ ď k and q ě 1, we have q Ñ8 x qk ` ℓ ď p k q x k ` ℓ x ℓ q{p q k ` ℓ q ď x k ` X k { q Ý Ý Ý Ñ x k and thus lim sup k Ñ8 x k ď lim inf k Ñ8 x k . Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of a finitely generated monoid In a monoid M generated by a finite family F , | τ | “ #generators (in F ) needed to write τ m k “ #elements of size k Lemma: m k ` ℓ ď m k m ℓ ( log m k is sub-additive ) Corollary: The sequence m 1 { k converges towards M ’s growth rate ! k Proof: If m k “ 0 for some k ě 0, then m ℓ “ 0 for all ℓ ě k . Otherwise, set x k “ p log m k q{ k ě 0 and X k “ max t x 1 , . . . , x k u . For all ℓ ď k and q ě 1, we have q Ñ8 x qk ` ℓ ď p k q x k ` ℓ x ℓ q{p q k ` ℓ q ď x k ` X k { q Ý Ý Ý Ñ x k and thus lim sup k Ñ8 x k ď lim inf k Ñ8 x k . Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of trace monoids F ` B ˇ S i S j “ S j S i ˇ T 4 “ S 1 , S 2 , S 3 , S 4 λ 4 , k “ # t τ P T 4 : | τ | “ k u ˇ if i ‰ j ˘ 1 ˇ How does λ 4 , k precisely behave when k Ñ `8 ? Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of trace monoids F ` B ˇ S i S j “ S j S i ˇ T 4 “ S 1 , S 2 , S 3 , S 4 λ 4 , k “ # t τ P T 4 : | τ | “ k u ˇ if i ‰ j ˘ 1 ˇ How does λ 4 , k precisely behave when k Ñ `8 ? Generating function λ 4 , k z k “ ÿ ÿ z | τ | G 4 p z q “ k ě 0 τ P T 4 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of trace monoids F ` B ˇ S i S j “ S j S i ˇ T 4 “ S 1 , S 2 , S 3 , S 4 λ 4 , k “ # t τ P T 4 : | τ | “ k u ˇ if i ‰ j ˘ 1 ˇ How does λ 4 , k precisely behave when k Ñ `8 ? Möbius polynomial Generating function P 4 p z q “ G 4 p z q ´ 1 “ 1 ´ 4 z ` 3 z 2 λ 4 , k z k “ ÿ ÿ z | τ | G 4 p z q “ k ě 0 τ P T 4 P 4 p z q 1 z 0 0 1 { 3 1 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of trace monoids F ` B ˇ S i S j “ S j S i ˇ T 4 “ S 1 , S 2 , S 3 , S 4 λ 4 , k “ # t τ P T 4 : | τ | “ k u ˇ if i ‰ j ˘ 1 ˇ How does λ 4 , k precisely behave when k Ñ `8 ? Möbius polynomial Generating function P 4 p z q “ G 4 p z q ´ 1 “ 1 ´ 4 z ` 3 z 2 λ 4 , k z k “ ÿ ÿ z | τ | G 4 p z q “ k ě 0 τ P T 4 P 4 p z q 1 ´ 1 ρ 4 “ 1 { 3 and λ 4 , k „ ρ k ` 1 z P 1 0 4 p ρ 4 q 4 0 1 { 3 1 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rate of trace monoids F ` B ˇ S i S j “ S j S i ˇ T 4 “ S 1 , S 2 , S 3 , S 4 λ 4 , k “ # t τ P T 4 : | τ | “ k u ˇ if i ‰ j ˘ 1 ˇ How does λ 4 , k precisely behave when k Ñ `8 ? Möbius polynomial Generating function P 4 p z q “ G 4 p z q ´ 1 “ 1 ´ 4 z ` 3 z 2 λ 4 , k z k “ ÿ ÿ z | τ | G 4 p z q “ k ě 0 τ P T 4 P 4 p z q 1 ´ 1 ρ 4 “ 1 { 3 and λ 4 , k „ ρ k ` 1 z P 1 0 4 p ρ 4 q 4 0 1 { 3 1 Corollary: λ 1 { k 4 , k Ñ 1 { ρ 4 “ 3 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rates of wide trace monoids How does ρ n behave when n Ñ `8 ? Recurrence equation Recurrence equation P n p z q 1 P ´ 1 p z q “ P 0 p z q “ 1 P n p z q “ P n ´ 1 p z q ´ zP n ´ 2 p z q z 0 ρ 5 ρ 4 ρ 3 ρ 2 ρ 1 if n ě 1 0 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rates of wide trace monoids How does ρ n behave when n Ñ `8 ? Recurrence equation Recurrence equation P n p z q 1 P ´ 1 p z q “ P 0 p z q “ 1 P n p z q “ P n ´ 1 p z q ´ zP n ´ 2 p z q z 0 ρ 5 ρ 4 ρ 3 ρ 2 ρ 1 if n ě 1 0 ρ n Ñ ρ 8 ě 0 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Growth rates of wide trace monoids How does ρ n behave when n Ñ `8 ? Recurrence equation Recurrence equation P n p z q 1 P ´ 1 p z q “ P 0 p z q “ 1 P n p z q “ P n ´ 1 p z q ´ zP n ´ 2 p z q z 0 ρ 5 ρ 4 ρ 3 ρ 2 ρ 1 if n ě 1 0 1 ρ n “ ρ n Ñ ρ 8 “ 1 { 4 4 cos p π n ` 2 q 2 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
Contents Growth rates of trace monoids 1 Growth rates of trace monoids: a first proof 2 Growth rates of braid monoids: a first proof 3 Growth rates of trace and braid monoids: an algebraic proof 4 Conclusion 5 Ramón Flores, Juan González-Meneses & Vincent Jugé Growth rates of wide braid monoids
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