based on similar combinatorial constructions. The proof that the WL-dimension is low intuitively indicates that the ability to fix products of elements appears to be related to the ability to fix sets of elements and how to exploit this.

In comparison to the previous constructions mentioned above, our construction has the advantage that it is of a purely combinatorial nature. It is therefore easy to analyze the groups, and many combinatorial properties of the resulting groups can be tuned. In fact, we can start with an arbitrary graph and encode it into a group while preserving isomorphisms. We should stress that even though we start with graphs of unbounded Weisfeiler-Leman dimension, the resulting groups have only dimension 3. This highlights the power of Weisfeiler-Leman-type algorithms to distinguish groups beyond the scope of traditional invariants.

Further related work
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Our work can be understood as studying the descriptive complexity of finite groups. We refer to Grohe’s monograph [@MR3729479] for extensive information on the descriptive complexity of graphs (rather than groups). A central result in [@MR3729479] shows graph classes with a forbidden minor have bounded WL-dimension. A recent paper relating first order logics and groups is [@MR3705849]. The descriptive complexity of finite 