小消去群和随机群的子复形共形维数
Conformal dimension via subcomplexes for small cancellation and random groups
摘要：
我们发现新的界限上的共形尺寸的小取消组。它们被用来证明一个随机的少关系群具有共形维数2+o（1）渐近几乎必然（a.a.s.）。事实上，如果关系子的数量在关系子的长度l上像l^K那样增长，那么a.a.s.这样一个随机群具有共形维数2+K+o（1）。在格罗莫夫的密度模型中，密度d<1/8 a.a.s.具有保形尺寸$\asymp dl / |\log d|$。C '（1/8）群的上界有两个主要成分：$\ell_p$ -上同调（遵循Bourdon-Kleiner）和Cayley复形中的壁（建立在Wise和Ollivier-Wise上）。为了找到下界，我们改进了[Mackay，2012]的方法，以在此类群体的Cayley复形中创建更大的“轮树”。作为推论，在d<1/8的密度模型中，密度d由边界的共形维数和群的欧拉示性数确定，直到一次幂。
英文摘要：
We find new bounds on the conformal dimension of small cancellation groups. These are used to show that a random few relator group has conformal dimension 2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of relators grows like l^K in the length l of the relators, then a.a.s. such a random group has conformal dimension 2+K+o(1). In Gromov's density model, a random group at density d<1/8 a.a.s. has conformal dimension $\asymp dl / |\log d|$. The upper bound for C'(1/8) groups has two main ingredients: $\ell_p$-cohomology (following Bourdon-Kleiner), and walls in the Cayley complex (building on Wise and Ollivier-Wise). To find lower bounds we refine the methods of [Mackay, 2012] to create larger `round trees' in the Cayley complex of such groups. As a corollary, in the density model at d<1/8, the density d is determined, up to a power, by the conformal dimension of the boundary and the Euler characteristic of the group.
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