论文标题
具有不同dehn功能的锥体等效nilpotent组
Cone-equivalent nilpotent groups with different Dehn functions
论文作者
论文摘要
对于每个$ k \ geqslant 3 $，我们都会展示一个简单连接的$ k $ -nilpotent lie lie $ n_k $，其dehn函数的行为与$ n^k $一样，而其相关carnot逐渐分级的$ \ mathsf {gr}（gr}（gr}（n_k）$的dehn函数）像$ n^{k+1} $。这种属性及其后果使我们能够揭示三个新现象。首先，由于这些组具有统一的晶格，因此提供了第一对有限呈现的组，具有bilipschitz渐近锥，但具有不同的dehn功能。这些组的第二个令人惊讶的功能是，对于每个均匀的整数$ k \ geqslant 4 $，$ n_k $的集中dehn函数的行为像$ n^{k-1} $，因此具有与dehn函数不同的指数。这回答了年轻的问题。最后，我们将注意力转移到了sublinear bilipschitz等价（SBE）上。由Cornulier引入的，这些图是诱导其渐近锥之间双裂吻合型同构的度量空间之间的地图。这些可以看作是准膜法的削弱，其中添加误差被均匀增长的功能$ v $取代。我们表明，$ v $ -sbe在$ n_k $和$ \ mathsf {gr}（n_k）$之间必须满足$ v（n）\ scucccurlyeq n^{1/（2k + 2）} $，从而增强了这两组并非排序的事实。这是为一对SBE组提供明确的下限的第一个实例。
For every $k\geqslant 3$, we exhibit a simply connected $k$-nilpotent Lie group $N_k$ whose Dehn function behaves like $n^k$, while the Dehn function of its associated Carnot graded group $\mathsf{gr}(N_k)$ behaves like $n^{k+1}$. This property and its consequences allow us to reveal three new phenomena. First, since those groups have uniform lattices, this provides the first examples of pairs of finitely presented groups with bilipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer $k \geqslant 4$ the centralized Dehn function of $N_k$ behaves like $n^{k-1}$ and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bilipschitz equivalences (SBE). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasiisometries where the additive error is replaced by a sublinearly growing function $v$. We show that a $v$-SBE between $N_k$ and $\mathsf{gr}(N_k)$ must satisfy $v(n)\succcurlyeq n^{1/(2k + 2)}$, strengthening the fact that those two groups are not quasiisometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.
