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In this paper we continue our study of hopficity begun in [1], [2], [3], [4] and [5]. Let$A$be hopfian and let$B$have a cyclic center of prime power order. We improve Theorem 4 of [2] by showing that if$B$has finitely many normal subgroups which form a chain (we say$B$is$n$-normal), then$AxB$is hopfian. We then consider the case when$B$is a$p$-group of nilpotency class 2 and show that in certain cases$AxB$is hopfian.

We investigate the operator -\Delta-lpha\delta (x-\Gamma) in -\Delta-lpha\delta (x-\Gamma) , where -\Delta-lpha\delta (x-\Gamma) is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion for the lower part of the spectrum as -\Delta-lpha\delta (x-\Gamma) which involves a``two-dimensional''comparison operator determined by the geometry of the surface -\Delta-lpha\delta (x-\Gamma) . In the compact case the asymptotics concerns negative eigenvalues, in the periodic case Floquet eigenvalues. We also give a bandwidth estimate in the case when a periodic -\Delta-lpha\delta (x-\Gamma) decomposes into compact connected components. Finally, we comment on analogous systems of lower dimension and other aspects of the problem.

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