Generators and Relations for Discrete Groups

Complete bipartite graphs with a unique regular embedding

“…Thus M i has type {2n, n} and genus (n − 1)(n − 2)/2, the exception for n = 4 being the map M 1 , which is the torus map {4, 4} 2,2 in the notation of Coxeter and Moser [4]. Similarly, one can show that if j is coprime to p then the mapping x → x j , y → y j induces an isomorphism G i → G ij , so the operation H j transforms M i to M ij .…”

“…Since xy = (h 1 h 2 , h 2 h 1 ) has order q, each map has type {2q, n} and genus 1 + n 2 (n − p − 2). For a given n these maps form a single orbit under the operations H j , where gcd(j, n) = 1; if q = 2 then the unique map M ( ∼ = {4, n} 4 in [4]) is reflexible, but otherwise they form φ(q)/2 chiral (mirror-image) pairs. where 0 i < p. This is an extension of a normal subgroup x, y p ∼ = C p e × C p e−1 by y / y p ∼ = C p , with y acting by x y = x p e−1 i+1 y −p e−1 i and (y p ) y = y p (an automorphism of order p), so |G| = p 2e .…”

“…(This embedding was originally constructed by Biggs and White [2, §5.6.7] as a Cayley map, and it also arises naturally as a dessin in the sense of Grothendieck on the nth degree Fermat curve: see Example 2 of Section 3.) The resulting map is regular; it has type {2n, n} in the notation of Coxeter and Moser [4], meaning that the faces are all 2n-gons and the vertices all have valency n, and it has genus (n − 1)(n − 2)/2.…”

Gaussian phase transitions and conic intrinsic volumes: Steining the Steiner formula

“…For the permutahedron, one can use Theorem 3 of [23, Theorem 3] (see also the first line of Table 10 of [18]) to deduce that…”

“…The calculation for the signed permutahedron is the analogous, but now one has to use [6, formula (3)]; see also the second line of Table 10 of [18].…”