Thompson群$F$的每个拷贝在$F$中都是无畸变的
Every copy of Thompson's group $F$ in $F$ is undistorted
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中文总结 AI 辅助
该研究否定回答了Thompson群$F$是否存在自身的畸变拷贝的问题，证明$F$中每个同构于$F$的子群都是无畸变的。
中文摘要 AI 辅助
Thompson群$F$是单位区间上所有分段线性同胚构成的群，其断点为二进数，斜率为2的整数次幂。若$H$是有限生成群$G$的有限生成子群，其畸变衡量$H$的内在字度量与由$G$诱导的度量之间的差异；当这些度量等价时，该子群是无畸变的。Guba和Sapir提出$F$是否包含自身的畸变拷贝的问题，Brin也提出了相同问题。我们否定地回答该问题：每个同构于$F$的$F$的子群在$F$中都是无畸变的。
英文摘要
Thompson's group $F$ is the group of all piecewise-linear homeomorphisms of the unit interval whose breakpoints are dyadic and whose slopes are integer powers of $2$. If $H$ is a finitely generated subgroup of a finitely generated group $G$, its distortion measures the difference between the intrinsic word metric of $H$ and the metric induced from $G$. The subgroup is undistorted when these metrics are equivalent. Guba and Sapir asked whether $F$ contains a distorted copy of itself. The same question was also suggested by Brin. We answer this question negatively: every subgroup of $F$ isomorphic to $F$ is undistorted in $F$.
