发表机构
Research Center in Mathematics and Applications (CIMA); Department of Mathematics, School of Sciences and Technology, University of Évora（数学与应用研究中心; 埃武拉大学理科学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了一个无限几乎自由群$G=C_4\ast_{C_2}D_3$，其Cayley图是2-测地但非测地，反驳了相关猜想，并证明其自由不可分解性。
AI 中文摘要
我们构造了一个显式的无限几乎自由群 $G=C_4\ast_{C_2}D_3$，它带有一个有限的反向封闭生成集，其Cayley图是2-测地的但不是测地的。更精确地，我们描述了测地线的完整语言，并证明了$G$的每个元素都有一个唯一的测地线代表，除了一个元素恰好有两个。这给出了一个反例，反驳了关于一个无限群若具有$k$-测地Cayley图则必须具有关于同一生成集的测地Cayley图的猜想。我们进一步证明了$G$是自由不可分解的，因此是非平凡的。因此，Shapiro关于每个测地群都是平凡的猜想以及每个双曲$k$-测地群都是测地群的猜想不能同时成立。
英文摘要
We construct an explicit infinite virtually free group $G=C_4\ast_{C_2}D_3$ with a finite inverse-closed generating set whose Cayley graph is $2$-geodetic but not geodetic. More precisely, we describe the full language of geodesics and show that every element of $G$ has a unique geodesic representative except for a single element, which has exactly two. This gives a counterexample to the conjecture that an infinite group admitting a $k$-geodetic Cayley graph must admit a geodetic Cayley graph with respect to the same generating set. We further prove that $G$ is freely indecomposable and hence non-plain. Consequently, Shapiro's conjecture that every geodetic group is plain and the conjecture that every hyperbolic $k$-geodetic group is geodetic cannot both hold.
Comments9 pages, comments are welcome
