We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

中文翻译：

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虚二次场的双曲镶嵌和生成器

我们开发了构造显式生成器模扭转的方法 $K_3$ - 二次虚数域组。这些方法基于双曲线的镶嵌 $3$ -空间或在合适的前布洛赫组中直接计算，并导致第一个证明的显式生成器示例，模扭转，任何无限 $K_3$ - 一组数字字段。作为这种方法的一部分，我们对布洛赫群的理论进行了一些改进 $ K_3 $ 任何领域，预测精确的力量 $2$ 这应该发生在 Lichtenbaum 猜想中 $ -1 $ 并证明这个预测对所有阿贝尔数域都有效。