On the genus of the semidirect product of ℤ9 by ℤ3
On the genus of the semidirect product of ℤ9 by ℤ3
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关于 ℤ9 与 ℤ3 的半直积的属
作者：
Matthew G. Brin;D. Rauschenberg;C. Squier
我们证明了，09与03的非交换半直积Γ有可定向亏格4。换句话说，Γ的某些凯莱图嵌入亏格为4的可定向曲面（欧拉特征数为−6），但没有Γ的凯莱图嵌入亏格小于4的可定向曲面（欧拉特征数大于−6）。我们还证明了Γ的某些Cayley图嵌入到Euler特征为−3的（不可定向的）曲面中，但没有Γ的Cayley图嵌入到Euler特征大于−3的曲面中。Γ是第一个已知的例子，一个群的可定向欧拉特征线和不可定向欧拉特征线相差大于1。我们的结果也完成了确定的可定向亏格的每一个组的阶小于32。
We show that the non-commutative semidirect product Γ of ℤ9 by ℤ3 has orientable genus 4. In other words, some Cayley graph of Γ embeds in an orientable surface of genus 4 (Euler characteristic −6), but no Cayley graph of Γ embeds in an orientable surface of genus less than 4 (Euler characteristic greater than −6). We also show that some Cayley graph of Γ embeds in a (non-orientable) surface of Euler characteristic −3, but no Cayley graph of Γ embeds in a surface of Euler characteristic greater than −3. Γ is the first known example of a group whose orientable Euler characteristic and non-orientable Euler characteristic differ by more than 1. Our results also complete the determination of the orientable genus of each group of order less than 32.