Rigidity theorems for non-arithmetic matrix groups

Rigidity theorems for non-arithmetic matrix groups

Rigidity theorems for non-arithmetic matrix groups
(算術的とは限らない行列群の剛性定理)
Masato MIMURA (見村 万佐人)
Graduate School of Mathematical Sciences,
University of Tokyo
要旨（English abstract below）
次の 2 つの群のクラスを考える：
(a) 高ランク格子 (各因子の局所ランクが 2 以上の半単純代数群 G の格子 Γ：G の離散部分
群で商空間 G/Γ に G のハール測度からの有限測度が誘導されるもの)．本講演では局所
体にアルキメデス的なものも含める；
(b) ランク 1 の格子 (SOn,1 , SUn,1 , Spn,1 の格子など) やグロモフの意味での双曲群．
このとき，以下のような現象が広く知られている：
(a) の群のほうが強い剛性をもち、(b) の群は相対的に “軟らかい”ふるまいをする．
本講演では，まず上で述べた剛性の具体例を挙げる．次に，高ランク格子 ((a) のクラスの群．
マルグリスの著名な結果より，算術的ないしは S-算術的であることが知られている) の剛性
を，一般の環上で定義された算術的とは限らない行列群に拡張した結果についてお話しした
い．具体的には，普遍格子 (universal lattice)；および，斜交普遍格子 (symplectic universal
lattice) と呼ばれる群についての剛性を紹介する．
以下，k は任意の正の整数を表すとする．普遍格子とは，整係数多項式環上の特殊線型
群 SLn (Z[x1 , . . . , xk ])，高ランク条件により n ≥ 3 を常に仮定する，の事を指す（Shalom,
Lubotzky による）．同様に斜交普遍格子とは，Sp2n (Z[x1 , . . . , xk ])，高ランク条件によりこ
のケースは n ≥ 2 を常に仮定する，の事を指す．これらの群を考える動機（“なぜ，高ラン
ク格子の非算術的拡張とみなせるのか”）は以下の通りである：高ランク格子の典型例とし
てには SLn (O), Sp2n (O)，O は体の整数環 (または S-整数環)，の形のものがある．SL3 (Z),
√ √
SL3 (Z[1/2]), Sp4 (Z[ 2, 3]), Sp4 (F2 [x]) など．考える群の剛性によっては商群に性質が遺
伝するので，もし普遍格子や斜交普遍格子でそのような剛性が示せれば，SLn (O), Sp2n (O)
の形の高ランク格子での性質のおおもととみなすことができる（同様の理由により，高ラン
クルート系に対応する多項式環上の基本シュバレー群も興味が持たれている）．さらに，普
遍格子；斜交普遍格子；および SL3 (Z[x, x−1 ]) などのそれらの商群として実現できる多くの
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群は高ランク格子（従って算術格子）として実現できないことに注意されたい．このことは，
高ランク格子が次の “マルグリス有限性”を満たすことに因る：
任意の正規部分群は，有限または指数有限である．
本講演で扱う剛性は以下のものである：
• ([Mim1], [Mim3]) Lp 空間，p ∈ (1, ∞)，上の等長表現を係数とする群の 1-コホモロ
ジーの消滅（[BFGM]．この性質は，Kazhdan の性質 (T) と呼ばれる剛性 ([Kaz]) を
さらに強めたものである）
；
• ([Mim1], [Mim3]) バナッハ空間上の等長表現係数の群の 2 次の有界コホモロジー
(bounded cohomology) の性質：通常のコホモロジーへの比較写像と呼ばれる自然な
写像 Hb2 (Γ; B, ρ) → H 2 (Γ; B, ρ) の単射性（この性質は，Monod の性質 (TT)([Mon])
と呼ばれる剛性と関連する）
；
• ([Mim2]) 擬準同型 (quasi-homomorphism, quasimorphism) の自明性；安定交換子長
(stable commutator length) の消滅（[Bav]．これらは自明実係数の 2 次の有界コホモ
ロジーと関連する）
；
• ([Mim3],[Mim1]) 次の群をターゲットとする準同型の超剛性 (準同型の像が常に有限で
あること)：(i) 円周上の (低階数の) 微分同相群；(ii) コンパクトで向きづけられた曲
面の写像類群；(iii) 自由群の (外部) 自己同型群．
具体的な定理の内容は下述の英文アブストラクトおよび，講演を参照されたい．
Abstract
Consider the following two classes of countable discrete groups: (a) higher rank lattices
(by this we mean discrete subgroups of finite covolume in semisimple algebraic groups over
local fields whose each factor has local rank at least 2); (b) rank 1 lattices (such as lattices
in SOn,1 , SUn,1 , or Spn,1 ), and Gromov hyperbolic groups (note that by a well-known
result of Margulis, all higher rank lattices are arithmetic (or S-arithmetic)). Then the
following phenomenon is widely known:
groups in class (a) is more “rigid”, and those in class (b) is relatively “softer”.
In this talk, first we provide examples of the “rigidity” above. Secondly, we generalize
them to the case of non-arithmetic matric groups over general rings. More precisely, we
treat the groups called universal lattices and symplectic universal lattices.
Let k be any positive integer. The universal lattice is the special linear group over the
polynomial ring over Z, SLn (Z[x1 , . . . , xk ]), where we assume n ≥ 3 as a higher rank
condition (this terminology is due to Shalom and Lubotzky). Similarly the symplectic
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universal lattice denotes the groups of the form Sp2n (Z[x1 , . . . , xk ]), n ≥ 2. One of the
motivations to consider them is the following: typical examples of higher rank lattices are
SLn (O), n ≥ 3; and Sp2n (O), n ≥ 2, where O is a ring of integers (or of S-integers), such
√ √
as: SLn (Z), SLn (Z[1/2]); and Sp2n (Z[ 2, 3]), Sp2n (F2 [x]). Certain types of rigidity pass
to group quotients. Therefore, it might be natural to ask whether universal lattices and
symplectic universal lattices are “mother groups” of such rigidity so that to the groups
mentioned in above this superrigidity is inherited. Note that universal and symplectic
universal lattices themselves; and many group quotients of those, such as SL3 (Z[x, x−1 ]),
cannot be realized as higher rank lattices (hence as arithmetic lattices), because they fail
to have the following property, which is called the Margulis Finiteness Property: every
normal subgroup is either finite or of finite index.
In this talk, we argue the following theorems concerning the rigidity as mentioned above:
Theorem A. ([Mim1]; [Mim3]) Let Γ be a finite index subgroup either of universal lattices
with n ≥ 4; or of symplectic universal lattices with n ≥ 3. Let p ∈ (1, ∞). Then Γ has
property (FLp ) of Bader–Furman–Gelander–Monod [BFGM]. Namely, for any isometric
linear Γ-representation ρ on any Lp space, H 1 (Γ; ρ) = 0.
Moreover Γ has property (FFLp )/T in the sense of [Mim1]. In particular, for any
isometric linear Γ-representation ρ on any Lp space with ρ 6⊇ 1Γ , the comparison map
Hb2 (Γ; ρ) → H 2 (Γ; ρ) is injective. Here Hb• denotes bounded cohomology (see [Mon]).
If p = 2 (in fact, if p ∈ (1, 2]), then (FLp ) is equivalent to Kazhdan’s property (T) [Kaz].
If p À 2, then (FLp ) is known to be strictly stronger than (T). Note that property (T) for
universal lattices; and generally, for elementary Chevalley groups of high rank is proven
[Sha]; [EJK]. Property (FLp ) for all p ∈ [2, ∞) implies the homomorphism superrigidity
1
into Diff 1+²
+ (S ) (namely, every homomorphism having finite image,) for every ² > 0.
Theorem B. ([Mim2]) Let A be a euclidean domain and Γ = SLn (A) (seen as a discrete
group). If n ≥ 6, then any quasi-homomorphism on Γ is bounded. In particular, if
A = K[x] where K is a field of infinite transcendence degree (such as C), then Γ satisfies
the following two properties:
(i) the stable commutator length, namely,
scl : [Γ, Γ] → R≥0 ; g 7→ lim n−1 · cl(g n )
n→∞
vanishes identically;
(ii) the commutator length cl is unbounded on [Γ, Γ](= Γ).
Here we say a map φ : G → R is a quasi-homomorphism if it satisies the homomorphism identity up to uniformly bounded error. In the theorem above, the first conclusion
implies (i) by Bavard’s duality theorem [Bav] (also it implies that the comparison map
Hb2 (Γ; R, 1Γ ) → H 2 (Γ; R, 1Γ ) is injective). Item (ii) follows from a result of van der Kallen.
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Theorem C. ([Mim3]) Let Γ be a finite index subgroup either of universal lattices or of
symplectic universal lattices. Then every homomorphism
Φ : Γ → MCG(Σg,l ) (g, l ≥ 0)
and every homomorphism
Ψ : Γ → Out(FN ) (N ≥ 2)
have finite image. Here MCG(Σg,l ) denotes the mapping class group of a compact oriented
surface with g genus and l punctures; and Out(FN ) denotes the outer automorphism group
of a free group of rank N .
Furthermore, we obtain the following result. In the statement below, En (A) denotes the
elementary group, namely, group generated by elementary matrices, over a ring A; and
Ep2n (A) denotes the elementary symplectic group:
Theorem D. ([Mim3]) Let Γ be a quotient group of a universal lattice or of a symplectic
universal lattice, such as En (A), n ≥ 3; or of Ep2n (A), n ≥ 2 for a finitely generated
commutative (unital and associative) ring A. Suppose a countable group Λ satisfies the
following two conditions:
(i) the group Λ is measure equivalent (see below) to Γ;
(ii) for an ergodic ME-coupling Ω of (Γ, Λ), there exists Λ-fundamental domain X ' Ω/Λ
in Ω such that the associated ME-cocycle α : Γ × X → Λ satisfies the L2 -condition.
Namely, for any γ ∈ Γ, |α(γ, ·)|Λ ∈ L2 (X). Here | · |Λ denotes a word metric on Λ
with respect to a finite generating set for Λ.
Then every homomorphism from Λ into MCG(Σg,l ) (g, l ≥ 0); or into Out(FN ) (N ≥ 2)
has finite image.
In particular, the following holds true: let Γ be a finite index subgroup of En (A), n ≥ 3;
or of Ep2n (A), n ≥ 2, where A is a finitely generated commutative (unital and associative)
ring. Suppose G be a locally compact and second countable group and G contains a group
isomorphic to Γ as a lattice. Then for any cocompact lattice Λ in G, every homomorphism
from Λ into MCG(Σg,l ) (g, l ≥ 0); or into Out(FN ) (N ≥ 2) has finite image.
Here two coutable groups are said to be measure equivalent (or shortly, ME ) if there
exists an infinite measure space (this measure space is called an ME coupling) with two
measurable, measure preserving actions assocaited with these groups such that these two
actions commute and that both actions have finite measure fundamental domains. Note
that by general theory of ergodicity, it is always possible to replace an ME coupling with
ergodic one. A basic and motivating case of ME groups is that two groups are lattices
in the same locally compact and second countable group. In this case, we can take this
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topological group as an measure space, and the corresponding two actions are respectively
the left multiplication and the right inverse multiplication. For an ME coupling Ω of two
ME groups Γ, Λ and a choice of a Λ-fundamental domain X (we regard X ∼
= Ω/Λ as a
Γ-space and write this Γ-action as γ · x to distinguish from the original Γ-action on Ω),
we can naturally define an ME cocycle α : Γ × X → Λ by the condition
α(γ, x)γx = γ · x.
For comprehensive treatment of ME, we refer to [Fur].
Theorem D generalizes homomorphism superrigidity for higher rank lattices into mapping class groups [FM]; and into outer automorphism groups of free groups [BW] to
nonarithmetic setting: not only from the case of SLn (O) and Sp2n (O), but also from the
case of any lattices of algebraic groups of the form SLn , Sp2n , or their products, including
cocompact lattices therein.
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