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 also 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.

The author acknowledges funding from the EU NextGenerationEU through the Recovery and Resilience Plan for Slovakia under the project No. 09I03-03-V04-00272.

The Nondeterministic Constraint Logic (NCL) framework (Hearn & Demaine; Theoret. Comput. Sci. 343(1-2); 2005) allows us to express any problem in PSPACE as the task of moving between fully or partially specified orientations of a simple cubic polyhedral graph having integral edge weights (of $1$ or $2$) and bounded bandwidth - i.e., a bounded gap between adjacent vertices in some linear ordering of the graph's vertex set - by way of sequentially reversing the orientations of directed edges under the constraint that all vertices have weighted in-degree at least $2$. Owing to its simple and intuitive nature, the NCL model has since become a cornerstone of PSPACE-completeness reductions for reconfiguration problems in graph theory, computational geometry, and formal models of puzzles and games.

In this talk, we discuss how the graph bandwidth constraint allowed for by the NCL model yields a natural encoding of NCL reachability problems as Wang tiling tasks for finite rectangular or toroidal areas. Concerning a specific application of this correspondence, we also show that the existence of linear time tiling algorithms for sequentially permissive (or ``brick'') Wang tile sets (Derouet-Jourdan, Mizoguchi, & Salvati; Mathematical Progress in Expressive Image Synthesis III. Mathematics for Industry 24; 2016), originally developed for modeling wall patterns, implies novel tractability results for the perfect matching reconfiguration problem (Bonamy et. al.; Proc. 44th MFCS; 2019) of finding restricted walks on the $1$-skeleton of the perfect matching polytope for a graph.

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.

The mix of two maniplexes is the minimal maniplex that covers both. This construction has many important applications, such as finding the smallest regular cover of a maniplex. If one of the maniplexes is an abstract polytope, a natural question to ask is whether the mix is also a polytope. We describe here a general criterion for the polytopality of the mix which generalizes several previously-known polytopality criteria.

Famously, every planar graph is $4$-colorable, and every triangle-free planar graph is $3$-colorable. Motivated by these intriguing facts, it is natural to ask what can be said about coloring of graphs that are "nearly planar", in the sense that they can be drawn in the plane with a bounded number of crossings. In this talk, I will survey results and open problems concerning this topic.

A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group theory, with applications extending to symmetric embedding of the complete bipartite graphs, such as regular dessins with underlying graphs $K_{m,n}$. A classical result by Douglas establishes that every bicyclic group is supersolvable. More recently, Fan and Li (2018) proved every finite $(m,n)$-bicyclic group is abelian if and only if $\gcd(m,\phi(n))=\gcd(n,\phi(m))=1$, where $\phi$ is Euler's totient function. In this talk we generalize this result further: we show that every $(m,n)$-bicyclic group is nilpotent if and only if $\gcd(n,\phi(\mathrm{rad}(m)))=\gcd(m,\phi(\mathrm{rad}(n)))=1$, where $\mathrm{rad}(m)$ denotes the radical of $m$ (the product of its distinct prime divisors).

Tilings have kindled fascinations across civilisations since antiquity. The studies of spherical tilings can be traced back to Plato. Regarded as spherical analogues of polyhedra, the five Platonic solids give rise to spherical tilings by congruent regular spherical polygons via projection. Beyond them, the quest to classify all the edge-to-edge tilings of the sphere by congruent polygons was pioneered by Duncan Sommerville in 1923. Euler's polyhedral formula asserts that the polygons are triangles, quadrilaterals, or pentagons. The tilings by congruent triangles were eventually classified by Yukako Ueno and Yoshio Agaoka in 2002. Significant progress has since been made in the directions of quadrilaterals and pentagons. The outstanding cases are nevertheless the most challenging and also of the utmost importance as they give rise to the majority of the tilings in question. We have recently solved all the outstanding cases and hence completed the whole classification celebrating the problem's centenary. The talk is based on joint works with Ho Man Cheung and Min Yan.

TBD

A classical result in the theory of maps on surfaces states that the medial of a regular map has two orbits on flags (e.g., it has two types of faces). If the original map is self-dual, however, its medial is regular — it admits symmetry that is somewhat unexpected. This prompts a natural analogue for truncation: what is the symmetry-type of the truncation of a given (say, regular) map? What conditions on the original map guarantee or forbid the emergence of unexpected symmetries for its truncation? Although the answer is known, it is less intuitive than in the medial case. In joint work with Hubard and Mochán, we introduced the notion of voltage operations, which generalise many classical operations on maps and polyhedra to the broader setting of maniplexes and abstract polytopes. In this talk, I will revisit voltage operations and present some results and ideas concerning the central question: when does an operation yield a result with unexpected symmetry?

In 2012 A. Blokhuis and A.E. Brower defined a graph on the flags of a biplane and proved that the graph corresponding to the unique $(11,5,2)$ is defined by its spectrum. We will look at these graphs for other biplanes, other graphs on flags of designs (symmetric and non-symmetric), some of their properties, and what we can say so far about their spectra.

This is a joint work with Octavio Baltasar Zapata-Fonseca.

Incidence geometries are in the basis of Tits buildings and related structures. Coset incidence systems are incidence structures derived from group cosets, where points, lines, and higher-dimensional elements correspond to cosets of certain subgroups. These capture symmetry and combinatorial properties of groups, particularly in relation to buildings and flag complexes. Given two groups $A$ and $B$, one can define different groups using group operations such as free products (with amalgamation), HNN-extensions, or split extensions. In this talk, given two coset incidence systems of the groups $A$ and $B$, we will describe two distinct ways of building a coset incidence system for its free product (with amalgamation), HNN-extension, and split extension. One of these constructions generalizes the twisting operation on polytopes [1], which is largely inspired on twisted simple groups.

References

1. McMullen, Peter, and Egon Schulte. Abstract regular polytopes. Vol. 92. Cambridge University Press, 2002.

A polycirculant drawing of a graph $G$ is a planar representation of $G$ that remains invariant under a non-trivial rotation, with each vertex orbit having the same size, denoted $m$. A graph $G$ paired with its semi-regular automorphism $a $ is termed a polycirculant structure $(G, a)$. The House of Graphs has recently introduced a new format for the input and output of graphs along with their vertex coordinates. This feature allows computer-generated drawings of graphs with precisely specified vertex coordinates which makes familiar graphs and graph families visually recognisable. In this talk, we focus on drawing polycirculant graphs, particularly those with a Hamiltonian cycle compatible with a given polycirculant structure. The presentation draws on my recent and ongoing work with coauthors, including Leah Berman, Simona Bonvicini, Gábor Gévay, Jan Goedgebeur, Gauvain Devillez and Arjana Žitnik. This research is supported in part by the Slovenian Research Agency through research program P1-0294 and projects J1-4351, J5-4596, and BI-HR/23-24-012.

Edge-to-edge tilings of the sphere by congruent quadrilaterals, where the quadrilaterals have straight edges, have been completely classified. We study the same problem, for the so-called curvilinear quadrilaterals, where the edges are not necessarily straight. Similar to the case of straight edges, there are four types of curvilinear quadrilaterals ($A^2BC$, $A^3B$, $A^2B^2$, $A^4$) that are suitable for tiling. In this talk, we classify tilings of the sphere by congruent curvilinear quadrilaterals of types $A^3B$ and $A^2BC$. For type $A^3B$, we use the "straightening" method to show the non-existence of some tilings that are combinatorially possible.

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.

TBD

In the classification of tilings of a surface by certain prototiles, the first step is usually the determination of all the possible combinations of the angles at vertices. For example, this is the first step in Rao’s classification of convex pentagons that can tile the plane. We discuss how this can be done for tilings of the sphere by angle congruent pentagons, and compare our approach to Rao's. Then we apply the results to get the classifications of some families of tilings of the sphere by angle congruent pentagons.

This is a joint work with Robert Barish and Hoi Ping Luk, and is supported by NSFC/RGC Joint Research Scheme N_HKUST607/23.