论文标题
双曲线组的度量结构空间
The space of metric structures on hyperbolic groups
论文作者
论文摘要
我们研究了空间的度量和拓扑特性$ \ mathscr {d}（g）$的左屈曲假伪图在非元素双曲线组$ g $上，这些$ g $是单词度量的准代表，直至一词，直至粗略相似。如果$ g $是表面组，那么当$ g $是一个免费组时，这个空间自然包含teichmüller空间。我们的自然公制让人联想到（对称的）瑟斯顿在teichmüller空间上的指标，我们证明$ \ mathscr {d}（g）$是一个无绑定的缩度度量空间，而$ \ sathrm {out}（out}（out}（g）$在ITSOMOMETER上可以在计算上适当地表现。如果我们将自己限制在子空间$ \ mathscr {d}_Δ（g）的$中，由$Δ$ - 氢 - 氢 - 氢 - 氢 - 氢 - 氢 - 氢含量1的指标1，我们证明它是空的或适当的。我们还从$ \ mathscr {d}_δ（g）$中证明了Bowen-Margulis映射的连续性，$ \ Mathbb {p} \ Mathcal {c} urr（g）uror（g）$ g $上的投射地球电流的$ g $，广泛的表面和自由组的相似结果，以及（正常的）功能（正常值）。 $ \ mathscr {d}（g）\ times \ times \ mathscr {d}（g）$。
We study the metric and topological properties of the space $\mathscr{D}(G)$ of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group $G$ that are quasi-isometric to a word metric, up to rough similarity. This space naturally contains the Teichmüller space in case $G$ is a surface group and the Culler-Vogtmann outer space when $G$ is a free group. Endowed with a natural metric reminiscent of the (symmetrized) Thurston's metric on Teichmüller space, we prove that $\mathscr{D}(G)$ is an unbounded contractible metric space and that $\mathrm{Out}(G)$ acts metrically properly by isometries on it. If we restrict ourselves to the subspace $\mathscr{D}_δ(G)$ of the points represented by $δ$-hyperbolic metrics with critical exponent 1, we prove that it is either empty or proper. We also prove continuity of the Bowen-Margulis map from $\mathscr{D}_δ(G)$ into the space $\mathbb{P}\mathcal{C}urr(G)$ of projective geodesic currents on $G$, extending similar results for surface and free groups, and the continuity of the (normalized) mean distortion as a function on $\mathscr{D}(G)\times \mathscr{D}(G)$.
