论文标题
Banach Property（t），用于$ \ rm sl_n（\ Mathbb {z}）$及其应用程序
Banach property (T) for $\rm SL_n (\mathbb{Z})$ and its applications
论文作者
论文摘要
我们证明，一个较高的较高等级的简单谎言组（包括$ n \ geq 3 $）的$ \ rm sl_n（\ mathbb {r}）$，他们的晶格就所有超级反省的Banach空间都具有Banach属性（T）。 该结果的两个后果是：首先，我们针对所有高级简单谎言组的大家族的所有超反射性Banach空间推断出Banach的固定点特性。例如，我们表明，对于每个$ n \ geq 4 $，$ \ rm sl_n（\ m马理{r}）$及其所有晶格都具有与所有超反射性Banach Space相对于所有超级反省的Banach Space。其次，我们解决了一个长期的开放问题，并表明Margulis Expanders（$ \ rm sl_ {n}的Cayley图形（\ Mathbb {Z} / M \ Mathbb {Z}）$ for固定$ n \ geq 3 $ 3 $和$ m $ tonding to Infinity to Infinity to to Infinity to 我们的所有结果源于以$ \ rm sl_3（\ mathbb {z}）$证明Banach属性（T）。我们的$ \ rm sl_3（\ mathbb {z}）$的证明方法依赖于$ \ rm sl_3（\ mathbb {z}）$的uni-Triangular子组的相对Banach属性（T）的新颖证明。即使在经典的希尔伯特环境中，这种相对属性（T）的证明也是新的，并且本身就很有趣。
We prove that a large family of higher rank simple Lie groups (including $\rm SL_n (\mathbb{R})$ for $n \geq 3$) and their lattices have Banach property (T) with respect to all super-reflexive Banach spaces. Two consequences of this result are: First, we deduce Banach fixed point properties with respect to all super-reflexive Banach spaces for a large family of higher rank simple Lie groups. For example, we show that for every $n \geq 4$, the group $\rm SL_n (\mathbb{R})$ and all its lattices have the Banach fixed point property with respect to all super-reflexive Banach spaces. Second, we settle a long standing open problem and show that the Margulis expanders (Cayley graphs of $\rm SL_{n} (\mathbb{Z} / m \mathbb{Z} )$ for a fixed $n \geq 3$ and $m$ tending to infinity) are super-expanders. All of our results stem from proving Banach property (T) for $\rm SL_3 (\mathbb{Z})$. Our method of proof for $\rm SL_3 (\mathbb{Z})$ relies on a novel proof for relative Banach property (T) for the uni-triangular subgroup of $\rm SL_3 (\mathbb{Z})$. This proof of relative property (T) is new even in the classical Hilbert setting and is interesting in its own right.
