The proposed project brings together Neha Nanda, a young researcher who finished her Ph.D this summer, and who already has a number of publications and pre-prints, and an experienced researcher, John Guaschi, who is full professor at Université de Caen Normandie. They propose to study certain ramifications of braid groups and related structures, namely the (virtual) twin groups, and the relations of these groups with (virtual) doodles that play the rôle of planar analogues of knots in R^3. The main aims of the project are: 1) understand better the (pure) twin groups by studying some of their invariants, such as their virtual cohomological dimension, and look for good presentations for them using Fadell-Neuwirth-like short exact sequences. 2) following the construction of surface braid groups from Artin braid groups, generalise the (pure) twin groups to surface (pure) twin groups, obtain presentations of these groups, and seek doodle-like structures to which they correspond. 3) construct new (virtual) doodle invariants, some being analogous to those known for knots and links, the aim being to classify (virtual) doodles. Being at the intersection of several domains, the general setting of the project is active and competitive. As well as being ambitious, the project is innovative, not just for the choice of problems to be studied, but also because its members have complementary interests and skills in algebra, low-dimensional topology, and combinatorial group theory. The project members will also be able to count on the expertise of other mathematicians in Caen working in related areas, as well as the dynamic network of the research group in Caen, including meetings organised by members of the GDR "Tresses", the regional ARTIQ project in collaboration with computer scientists in Rouen, and schools such as "Winterbraids".