On the number of regions formed by arrangements of closed geodesics on flat surfaces

Face enumeration for line arrangements in a 2-torus

“…The same formula was also independently discovered by Lawrence in [6] and by the second author in [2]. Recently, Shnurnikov has characterized the set of all possible values of f 2 for toric line arrangements in [11]. See also [12] for arrangements in hyperbolic spaces, icosahedron and also arrangements of immersed circles in surfaces.…”

“…From left: toric arrangements with f -vectors (4,8,4), (4,9,5) and (4,10,6) Figure 6. From left: toric arrangements with f -vectors (4,11,7) and (4,12,8) Combining above inequality with Theorem 2.7 we see that for an arrangement of n toric lines f 0 ∈ { n−1 2 } ∪ {l : l ≥ n − 2}. As f 0 is not bounded above there is no hope for a complete characterization of the pairs (n, f 0 ).…”

“…The Smith normal formal form of a 2×2 matrix A is a diagonal matrix with 1 in the (1, 1) position and det A in (2, 2) position. The subgroup of the torus corresponding to A is then a finite abelian group of order | det A| as the normal form is obtained by unimodular transformations (see [11,Lemma 1] for a geometric proof). As before we will denote A by an augmented matrix [A | c].…”

“…The construction is illustrated in Figure 4 for n = 6. (6,9), (6,10) and (6,11) In order to prove the reverse containment assume that A contains at most m toric lines of the same type (i.e., m is the maximal number of parallel lines). If m = n − 1 then f 2 is a multiple of n − 1.…”

Face counting formula for toric arrangements defined by root systems

“…The formula for the number of chambers for such arrangements was first discovered by Ehrenborg et al in [6]; it also appears in [7] and [8]. Recently, Shnurnikov has characterized the set of all possible values of top-dimensional faces for arrangements in a 2-torus in [9] and for arrangements in higher-dimensional tori in [10]. See also [11] for a complete characterization of face numbers in case of arrangements in a 2-torus.…”