Schottky space S_g , where g ⩾ 2 is an integer, is a connected complex orbifold of dimension 3(g − 1); it provides a parametrization of the PSL₂ (ℂ)-conjugacy classes of Schottky groups Γ of rank g. The branch locus B_g ⊂ S_g, consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian group, is known to be connected. If [Γ] ∈ Bg , then there is a Kleinian group 𝐾 containing Γ as a normal subgroup of index some prime integer 𝑝 ⩾ 2. The structural description, in terms of Klein–Maskit Combination Theorems, of such a group 𝐾 is completely determined by a triple (𝑡, 𝑟, 𝑠), where 𝑡, 𝑟, 𝑠 ⩾ 0 are integers such that g = 𝑝(𝑡 + 𝑟 + 𝑠 − 1) + 1 − 𝑟. Foreach such tuple (g, 𝑝; 𝑡, 𝑟, 𝑠), there is a corresponding cyclic-Schottky stratum 𝐹(g, 𝑝; 𝑡, 𝑟, 𝑠) ⊂ Bg . It is known that 𝐹(g, 2; 𝑡, 𝑟, 𝑠) is connected. In this paper, for 𝑝 ⩾ 3, we study the connectivity of these 𝐹(g, 𝑝; 𝑡, 𝑟, 𝑠).

The complex orbifold structure of the moduli space of Riemann surfaces of genus g (g≥2) produces a stratification into complex subvarieties named equisymmetric strata. Each equisymmetric stratum is formed by the surfaces where the group of automorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata of dimension one are of two structurally different types. Type 1 equisymmetric strata correspond to Riemann surfaces where the group of automorphisms produces a quotient surface of genus zero, while those of Type 2 appear when such a quotient is a surface of genus one. Type 1 equisymmetric strata have been extensively studied by the authors of the present work in a previous recent paper, we now focus on Type 2 strata. We first establish the existence of such strata and their frequency of occurrence in moduli spaces. As a main result we obtain a complete description of Type 2 strata as coverings of the sphere branched over three points (Belyi curves) and where certain isolated points (punctures) have to be eliminated. Finally, we study in detail the doubly infinite family of Type 2 strata whose automorphism groups have order the product of two primes.

We classify compact Riemann surfaces of genus g, where g−1 is a prime p, which have a group of automorphisms of order ρ(g−1)for some integer ρ≥1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ>6, and of the first and third authors for ρ= 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p+1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps.

In this article we determine the maximal possible order of the automorphism group 18 of the form ag + b, where a and b are integers, of a complex three and four- 19 dimensional family of compact Riemann surfaces of genus g, appearing for all genus.In addition, we construct and describe explicit complex three and four-dimensional 20 families possessing these maximal numbers of automorphisms.

We classify compact Riemann surfaces of genus g, where _g−1 is a prime p, whichhave a group of automorphisms of order ρ(g−1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3,4,5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p +1, together with the non-orientable regular hypermaps of characteristic − p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps.

Unfortunatelly an erratum appears in the foreword of this volume.

We claim that one of the authors of the article On universal groups and three-manifolds, Invent. Math. 87 (1987), no. 3, 441–456 is W. Witten. The correct name of this author is Wilbur Carrington Whitten. W. C. Whitten is not the father of the fields medallist E. Witten, as we claim there.

Schottky space Sg is the space that parametrizes PSL2(C)-conjugacy classes of Schottky groups of rank g ≥ 2. The branch locus Bg consists of the conjugacy classes of those Schottky groups which are a finite index proper subgroup of some Kleinian group. In a previous paper we observed that Bg was connected for g ≥ 3 odd and that it has at most two components for g ≥ 4 even. In this short note, we observe that Bg is always connected.

We prove that themaximal number ag+b of automorphisms of equisymmetric and

complex-uniparametric families of Riemann surfaces appearing in all genera is 4g + 4. For

each integer g ≥ 2 we find an equisymmetric complex-uniparametric family Ag of Riemann

surfaces of genus g having automorphism group of order 4g + 4. For g ≡ −1mod 4 we

present another uniparametric family Kg with automorphism group of order 4g + 4. The

family Ag contains the Accola–Maclachlan surface and the family Kg contains the Kulkarni

surface

María Teresa, Maite, Lozano is a great person and mathematician, in these pages we can only give a very small account of her results trying to resemble her personality. We will focus our attention only on a few of the facets of her work, mainly in collaboration with Mike Hilden and José María Montesinos because as Maite Lozano pointed out in an international conference in Umeå University in June 2017, where she was a plenary speaker:

I am specially proud of been part of the team Hilden-Lozano-Montesinos (H-L-M), and of our mathematical achievements