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.
[1]
Viera Krnanová Proulx.
Classification of the toroidal groups
,
1978,
J. Graph Theory.
[2]
Thomas W. Tucker.
A refined Hurwitz theorem for imbeddings of irredundant Cayley graphs
,
1984,
J. Comb. Theory, Ser. B.
[3]
Thomas W. Tucker.
Some results on the genus of a group
,
1981,
J. Graph Theory.
[4]
H. Coxeter,et al.
Generators and relations for discrete groups
,
1957
.
[5]
Thomas W. Tucker,et al.
Finite groups acting on surfaces and the genus of a group
,
1983,
J. Comb. Theory, Ser. B.
[6]
M. Hall.
The Theory Of Groups
,
1959
.
[7]
Craig C. Squier,et al.
On the Genus of Z3 × Z3 × Z3
,
1988,
Eur. J. Comb..