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We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.
It is known that every finite group can be represented as the full group of automorphisms of a suitable compact dessin denfant. In this paper, we give a constructive and easy proof that the same holds for any countable group by considering non-compac
A dessin denfant, or dessin, is a bicolored graph embedded into a Riemann surface, and the monodromy group is an algebraic invariant of the dessin generated by rotations of edges about black and white vertices. A rational billiards surface is a two d
We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a si
Let $n, k geq 3$. In this paper, we analyse the quotient group $B_n/Gamma_k(P_n)$ of the Artin braid group $B_n$ by the subgroup $Gamma_k(P_n)$ belonging to the lower central series of the Artin pure braid group $P_n$. We prove that it is an almost-c
We prove that an analogue of Jordans theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of Vladimir L. Popov.