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This article deals with the study of cactus groups from a combinatorial pointof view. These groups have been gaining prominence lately in various domains ofmathematics, amongst which are their relations with well-known groups such asbraid groups, diagram groups, to name a few. We compute a minimal presentationfor cactus groups in terms of generators and non-redundant relations. We alsoconstruct homomorphisms of these groups onto certain finite groups, which leadsto results about finite quotients of cactus groups. More precisely, we provethat all (infinite) dihedral groups appear as quotients of cactus groups. Wealso investigate the lower central series and its consecutive quotients. Whilethere are already known established similarities with braid groups, we deduce aconsiderable disparity between the two groups.

Minimal presentation, finite quotients and lower central series of cactus groups