We give a systematic way to construct almost conjugate pairs of finite subgroups of $$\mathrm {Spin}(2n+1)$$Spin(2n+1) and $${{\mathrm{Pin}}}(n)$$Pin(n) for $$n\in {\mathbb {N}}$$n∈N sufficiently large. As a geometric application, we give an infinite family of pairs $$M_1^{d_n}$$M1dn and $$M_2^{d_n}$$M2dn of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions $$d_n>6$$dn>6. We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.

中文翻译：

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等谱近 Kähler 流形

我们给出了一种系统的方法来构造 $$\mathrm {Spin}(2n+1)$$Spin(2n+1) 和 $${{\mathrm{Pin}}}(n)$ 的有限子群的几乎共轭对$Pin(n) 对于 $$n\in {\mathbb {N}}$$n∈N 足够大。作为一个几何应用，我们给出了近 Kähler 流形的无限对 $$M_1^{d_n}$$M1dn 和 $$M_2^{d_n}$$M2dn，这些流形对于 Dirac 和 Laplace 算子具有增加的维度 $ $d_n>6$$dn>6。我们还提供了（局部）齐次六维近 Kähler 流形的体积计算，并研究了该维度中 Sunada 对的存在。