On the genus of the semidirect product of ℤ9 by ℤ3

✍ Scribed by Matthew G. Brin; David E. Rauschenberg; Craig C. Squier

• Book ID: 102892536
• Publisher: John Wiley and Sons
• Year: 1989
• Tongue: English
• Weight: 632 KB
• Volume: 13
• Category: Article
• ISSN: 0364-9024
• DOI: 10.1002/jgt.3190130108

✦ Synopsis

We show that the non-commutative semidirect product r of Z, by Z, has orientable genus 4. In other words, some Cayley graph of r embeds in an orientable surface of genus 4 (Euler characteristic -61, but no Cayley graph of r embeds in an orientable surface of genus less than 4 (Euler characteristic greater than -6). We also show that some Cayley graph of r embeds in a (non-orientable) surface of Euler characteristic -3, but no Cayley graph of r embeds in a surface of Euler characteristic greater than -3. r 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.