Collaborative perception, aiming at achieving a comprehensive perception range through inter-agent communication, faces challenges such as high communication costs and domain gaps between multiple agents. This paper introduces SparseComm, an innovative sparse communication collaborative perception framework designed to mitigate these challenges. SparseComm efficiently operates in sparse feature spaces and aggregates features related to the same objects by a sparse instance communication module. Meanwhile, a sparse 3D cooperation module is incorporated to enhance 3D feature representation during communication, thus improving detection performance. Furthermore, a bounding box restoration module is designed to recover undetected bounding boxes due to feature fusion and to address the quality drop issue caused by domain gaps at minimal additional communication cost. Extensive experiments conducted on the DAIR-V2X and V2XSet demonstrate the efficacy of SparseComm, achieving 47.12% at 211.46 communication bytes on DAIR-V2X and 78.03% at 216.63 communication bytes on V2XSet. Notably, SparseComm reduces communication consumption from 10× to 1000× compared with the prior methods while maintaining the detection performance.