Abstracts

A skew morphism of a finite group $G$ is an element $\varphi$ of $\textrm{Sym}(G)$ preserving the identity element of $G$ and having the property that for each $a\in G$ there exists a non-negative integer $i_a$ such that $\varphi(ab)=\varphi(a)\varphi^{i_a}(b)$ for all $b\in G$. Skew morphisms play an important role in the study of regular Cayley maps. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available to date. In this talk we show that if a skew morphism $\varphi$ of $\mathbb{Z}_n$ is not an automorphism of $\mathbb{Z}_n$, then it is uniquely determined by an element $h$ of $\mathbb{Z}_n$, a skew morphism $\alpha$ of $\mathbb{Z}_a$ where $a<n$, and a skew morphism $\beta$ of $\mathbb{Z}_b$ where either $b<n$, or $b=n$ and $\textrm{ord}(\beta)<\textrm{ord}(\varphi)$. Conversely, we a lso list necessary and sufficient conditions for a triple $(h,\alpha,\beta)$ to define a skew morphism of a given cyclic group. In particular, this gives a recursive characterisation of skew morphisms of finite cyclic groups.

In this talk I'll describe some recent discoveries (since 2022) about regular maps of small genus (in joint work with Primož Potočnik) and about finite quotients of triangle groups (in joint work with PhD student Darius Young).

As is well known, (ordinary) triangle groups of the form \(\Delta^+(k,l,m) = \langle\, x,y,z\ |\ x^k = y^l = z^m = xyz = 1\,\rangle\) (Equation) play an important role in the study of large automorphism groups of algebraic curves and compact Riemann surfaces, and of regular maps on orientable and non-orientable surfaces.

Much of the early part of the study of these things (dating back over 100 years) considered only small quotients of triangle groups, and subsequent work concentrated on finite simple quotients. But surprisingly, the recent determination of all orientably-regular maps of genus up to 1501 (by MC & PP) has shown that finite simple groups and finite insoluble groups account respectively for less than 0.1% and less than 7% of the associated quotients of ordinary triangle groups, while finite soluble quotients account for over 93%. Some other information is also interesting, such as the increasing proportion of orientably-regular maps of genus $2$ to given $g$ that are chiral. The long-standing question of what happens asymptotically (as $g \to \infty$) remains open.

Very little attention has been paid to soluble quotients, and as one step towards correcting this, a new theorem (by MC & DY, using a 1970 theorem by David Singerman) shows how every non-perfect hyperbolic ordinary triangle group has a smooth finite soluble quotient with derived length $c$ for some $c \le 3$, and has infinitely many such quotients with derived length $d$ for every $d > c$.

Also I will report on some work by Darius in 2024 in which he proved that the natural density (in the positive integers) of the set of orders of finite quotients of a triangle group $\Delta^+(k,l,m)$ is zero for every triple $(k,l,m)$. This work answers a question raised by Tom Tucker (who showed it holds when one of $k,l,m$ is coprime to the other two), and also completes and considerably extends an initial investigation by Larsen (2001), but avoids resorting to Larsen's use of the classification of finite simple groups.

Incidentally, it has also led to unexpected discoveries (by MC & Gabriel Verret & DY) about the densities of finite quotients of free products of two cyclic groups, and hence to the densities of orders of automorphism groups of regular maps of type $\{m,k\}$ for fixed $k$ and variable $q$ (or vice versa), and the densities of orders of finite locally-transitive graphs, and surprisingly, also about possibilities for the density of orders of finite quotients of a given finitely-generated group.

A maniplex of rank $n$ is an $n$-valent properly edge-coloured graph that generalises, simultaneously, maps on surfaces and abstract polytopes.he T problem of stability in maniplexes is a natural variant of the problem of stability in graphs in general. A maniplex is stable if every automorphism of its canonical double cover is a lift of some automorphism of the original maniplex. Due to their very rich structure, regular (maximally symmetric) maniplexes are always stable. It is thus natural to ask what is the maximum possible degree of symmetry that a maniplex that is not stable can admit.