单关系子群的单纯体积与稳定换位子长
Simplicial volume of one-relator groups and stable commutator length
摘要：
一个单关系者群是一个允许具有单个关系$r$的表示$\langle S \mid r \rangle$的群$G_r$。单关系子群在几何群论中形成了一类丰富的经典研究群。如果$r \in F(S)'$是$F(S)$的换位子群，则引入$\| G_r \|$的单纯体积。我们把这个不变量与元素$r \in F(S)$的稳定换位子长度$\textrm{scl}_S(r)$联系起来。我们表明，经常（但不总是）的线性关系$\| G_r \| = 4 \cdot \textrm{scl}_S(r) - 2$举行，每一个有理数模$1$是一个单关系组的单纯体积。此外，我们表明，这种关系保持适当的权力和满足小取消条件$C'(1/N)$的元素，与$O(1/N)$的乘法误差。这使我们能够证明随机元素的$F(S)'$的长度$n$的$\| G_r \|$是$2 \log(2 |S| - 1)/3 \cdot n / \log(n) + o(n/\log(n))$具有很高的概率，使用类似的结果Calegari-Walker稳定的换向器长度。
英文摘要：
A one-relator group is a group $G_r$ that admits a presentation $\langle S \mid r \rangle$ with a single relation $r$. One-relator groups form a rich classically studied class of groups in Geometric Group Theory. If $r \in F(S)'$, the commutator subgroup of $F(S)$, we introduce the simplicial volume of $\| G_r \|$. We relate this invariant to the stable commutator length $\textrm{scl}_S(r)$ of the element $r \in F(S)$. We show that often (though not always) the linear relationship $\| G_r \| = 4 \cdot \textrm{scl}_S(r) - 2$ holds and that every rational number modulo $1$ is the simplicial volume of a one-relator group. Moreover, we show that this relationship holds approximately for proper powers and for elements satisfying the small cancellation condition $C'(1/N)$, with a multiplicative error of $O(1/N)$. This allows us to prove for random elements of $F(S)'$ of length $n$ that $\| G_r \|$ is $2 \log(2 |S| - 1)/3 \cdot n / \log(n) + o(n/\log(n))$ with high probability, using an analogous result of Calegari-Walker for stable commutator length.
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