Existing AI-based point cloud compression methods struggle with dependence on specific training data distributions, which limits their real-world deployment. Implicit Neural Representation (INR) methods solve the above problem by encoding overfitted network parameters to the bitstream, resulting in more distribution-agnostic results. However, due to the limitation of encoding time and decoder size, current INR based methods only consider lossy geometry compression. In this paper, we propose the first lossless point cloud geometry compression method called Lossless Implicit Neural Representations for Point Cloud Geometry Compression (LINR-PCGC). To accelerate encoding speed, we design a group of point clouds level coding framework with an effective network initialization strategy, which can reduce around 60% encoding time. A lightweight coding network based on multiscale SparseConv, consisting of scale context extraction, child node prediction, and model compression modules, is proposed to realize fast inference and compact decoder size. Experimental results show that our method consistently outperforms traditional and AI-based methods: for example, with the convergence time in the MVUB dataset, our method reduces the bitstream by approximately 21.21% compared to G-PCC TMC13v23 and 21.95% compared to SparsePCGC.