Arithmetic quotients of the mapping class group
Arithmetic quotients of the mapping class group
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映射类组的算术商
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发表时间：
2013
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通讯作者：
Justin Malestein
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作者：
F. Grunewald;M. Larsen;A. Lubotzky;Justin Malestein
对于有限组H的每一个$$ {\ Mathbb {q}} $ q- mirredible代表r，与$$的简单因素a的简单因素a {\ mathbb {q} [h]} $ q [h] $ q [h] $$ {\ tau} $$τ。 $$ {（2G-2）\ times（2G-2）} $$（2G-2）×（2G-2）矩阵在自然顺序上$$ {\ mathfrak {o}^{op}} $$ op of $$ {a^{op}} $$ aop aop aop ap ap a^{a^{2g-2}}}上保留天然偏斜的sesquilinear形式$$ a2g-2。 } $$ mod（σg），即$$ {\ omega} $$ω的有限索引子组是有限索引的报价$$ {{\ rm mod}的子组（\ sigma_g）} $$ mod（σg）。商（和“ Torelli子组”），经典商$$ {{{\ rm sp}（2G，\ Mathbb {z}）} $$ sp（2g，z）只是列表中的第一种情况，对应于琐碎的h和琐碎的h和r。 $$ {{{\ rm mod}（\ sigma_g）} $$ mod（σg）的新算术报价，这些引号（\ sigma_g）} $$ mod（σg）定义在各种（子字段）的环形磁场（子字段）上，并具有$ $ {{\ rm sp}（2m），2m），， {\ rm so}（2M，2M），} $$ SP（2M），SO（2M，2M）和$$ {{\ rm Su}（M，M，M）} $$ SU（M，M）任意大m。
To every $${\mathbb{Q}}$$Q-irreducible representation r of a finite group H, there corresponds a simple factor A of $${\mathbb{Q}[H]}$$Q[H] with an involution $${\tau}$$τ. To this pair $${(A, \tau)}$$(A,τ), we associate an arithmetic group $${\Omega}$$Ω consisting of all $${(2g-2) \times (2g-2)}$$(2g-2)×(2g-2) matrices over a natural order $${\mathfrak{O}^{op}}$$Oop of $${A^{op}}$$Aop which preserve a natural skew-Hermitian sesquilinear form on $${A^{2g-2}}$$A2g-2. We show that if H is generated by less than g elements, then $${\Omega}$$Ω is a virtual quotient of the mapping class group $${{\rm Mod}(\Sigma_g)}$$Mod(Σg), i.e. a finite index subgroup of $${\Omega}$$Ω is a quotient of a finite index subgroup of $${{\rm Mod}(\Sigma_g)}$$Mod(Σg). This shows that $${{\rm Mod}(\Sigma_g)}$$Mod(Σg) has a rich family of arithmetic quotients (and “Torelli subgroups”) for which the classical quotient $${{\rm Sp}(2g, \mathbb{Z})}$$Sp(2g,Z) is just a first case in a list, the case corresponding to the trivial group H and the trivial representation. Other pairs of H and r give rise to many new arithmetic quotients of $${{\rm Mod}(\Sigma_g)}$$Mod(Σg) which are defined over various (subfields of) cyclotomic fields and are of type $${{\rm Sp}(2m), {\rm SO}(2m, 2m),}$$Sp(2m),SO(2m,2m), and $${{\rm SU}(m, m)}$$SU(m,m) for arbitrarily large m.