Regular Automorphism Research Articles

No bounded geometry wandering domains for sufficiently regular automorphisms

A question whether sufficiently regular manifold automorphisms may have wandering domains with controlled geometry is answered in the negative for quasiconformal or smooth homeomorphisms of n n -tori, n ≥ 2 n\ge 2 , and hyperbolic surfaces. Besides control on geometry of wandering domains, the assumptions are either analytic, e.g., minimal sets having measure zero or supporting invariant conformal structures, or geometric, such as uniform relative separation of wandering domains.

Regular automorphisms and Calogero-Moser families

Regular automorphisms and Calogero-Moser families

Genuinely nonabelian partial difference sets

AbstractStrongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

On Extension of a Regular Automorphism

On Extension of a Regular Automorphism

WHEN IS THE AUTOMORPHISM GROUP OF AN AFFINE VARIETY NESTED?

For an affine algebraic variety X, we study the subgroup Autalg(X) of the group of regular automorphisms Aut(X) of X generated by all the connected algebraic subgroups. We prove that Autalg(X) is nested, i.e., is a direct limit of algebraic subgroups of Aut(X), if and only if all the 𝔾a-actions on X commute. Moreover, we describe the structure of such a group Autalg(X).

A duality of scaffolds for translation association schemes

Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished “root” nodes and with matrix edge weights, often taken from Bose–Mesner algebras. In this paper, we first present a slight modification of Martin's conjecture (2021) concerning a duality of scaffolds whose digraphs are embedded in a closed disk in the plane with root nodes all lying on the boundary circle, and then show that this modified conjecture holds true if we restrict ourselves to the class of translation association schemes, i.e., those association schemes that admit abelian regular automorphism groups.

Characterization of affine surfaces with a torus action by their automorphism groups

In this paper we prove that if two normal affine surfaces $S$ and $S'$ have isomorphic automophism groups, then every connected algebraic group acting regularly and faithfully on $S$ acts also regularly and faithfully on $S'$. Moreover, if $S$ is non-toric, we show that the dynamical type of a 1-torus action is preserved in presence of an additive group action. We also show that complex affine toric surfaces are determined by the abstract group structure of their regular automorphism groups in the category of complex normal affine surfaces using properties of the Cremona group. As a generalization to arbitrary dimensions, we show that complex affine toric varieties, with the exception of the algebraic torus, are uniquely determined in the category of complex affine normal varieties by their automorphism groups seen as ind-groups.

On Periodic Groups with a Regular Automorphism of Order 4

On Periodic Groups with a Regular Automorphism of Order 4

Pentagonal quasigroups, their translatability and parastrophes

Abstract Any pentagonal quasigroup Q Q is proved to have the product x y = φ ( x ) + y − φ ( y ) xy=\varphi \left(x)+y-\varphi (y) , where ( Q , + ) \left(Q,+) is an Abelian group, φ \varphi is its regular automorphism satisfying φ 4 − φ 3 + φ 2 − φ + ε = 0 {\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε \varepsilon is the identity mapping. All Abelian groups of order n < 100 n\lt 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity ( x y ⋅ x ) y ⋅ x = y \left(xy\cdot x)y\cdot x=y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is { 1 1 n : n = 0 , 1 , 2 , … } \left\{1{1}^{n}:n=0,1,2,\ldots \right\} . We prove that the only translatable commutative pentagonal quasigroup is x y = ( 6 x + 6 y ) ( mod 11 ) xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}11) . The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group Z n {{\mathbb{Z}}}_{n} and its automorphism φ ( x ) = a x \varphi \left(x)=ax is proved to determine the value of a a and the range of values of n n .

Automorphism Group and Other Properties of Zero Component Graph over a Vector Space

In this paper, we introduce an undirected simple graph, called the zero component graph on finite-dimensional vector spaces. It is shown that two finite-dimensional vector spaces are isomorphic if and only if their zero component graphs are isomorphic, and any automorphism of a zero component graph can be uniquely decomposed into the product of a permutation automorphism and a regular automorphism. Moreover, we find the dominating number, as well as the independent number, and characterize the minimum independent dominating sets, maximum independent sets, and planarity of the graph. In the case that base fields are finite, we calculate the fixing number and metric dimension of the zero component graphs and determine vector spaces whose zero component graphs are Hamiltonian.

Abelian difference sets with the symmetric difference property

A $$(v,k,\lambda )$$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. The designs associated to the symplectic difference sets introduced by Kantor (J Algebra 33:43–58, 1975) have the SDP. Parker (J Comb Theory Ser A 67:23–43, 1994) claimed that the symplectic design on 64 points is the only SDP design on 64 points admitting an abelian regular automorphism group (an abelian difference set). We show in this paper that there is an SDP design on 64 points that is not isomorphic to the symplectic design and yet admits the group $$C_8 \times C_4 \times C_2$$ as a regular automorphism group. This abelian difference set is the first in an infinite family of abelian difference sets whose designs have the SDP and yet are not isomorphic to the symplectic designs of the same order. We define a new method for establishing the non-isomorphism of the two families.

The point regular automorphism groups of the Payne derived quadrangle of W(q)

In this paper, we completely determine the point regular automorphism groups of the Payne derived quadrangle of the symplectic quadrangle W(q), q odd. As a corollary, we show that the finite groups that act regularly on the points of a finite generalized quadrangle can have unbounded nilpotency class.

Finite 3-Subgroups in the Cremona Group of Rank 3

We consider 3-subgroups in groups of birational automorphisms of rationally connected threefolds and show that any 3-subgroup can be generated by at most five elements. Moreover, we study groups of regular automorphisms of terminal Fano threefolds and prove that, in all cases which are not among several explicitly described exceptions any 3-subgroup of such group can be generated by at most four elements.

Regular polynomial automorphisms in the space of planar quadratic rational maps

In this paper, we describe the semistable quotient of the set of regular polynomial automorphisms $${\mathcal {H}}_2^2$$ in the semistable locus of the moduli space of quadratic rational maps, using the portrait moduli space of rational maps with a fixed point. We also provide the parametrization of $${\mathcal {H}}_2^2$$ using two invariants, $$\det f$$ and $${\text {tr}}\,f$$ .

Braid group symmetries of Grassmannian cluster algebras

Let $${{\,\mathrm{\text{ Gr }}\,}}^\circ (k,n) \subset {{\,\mathrm{\text{ Gr }}\,}}(k,n)$$ denote the open positroid stratum in the Grassmannian. We define an action of the extended affine d-strand braid group on $${{\,\mathrm{\text{ Gr }}\,}}^\circ (k,n)$$ by regular automorphisms, for d the greatest common divisor of k and n. The action is by quasi-automorphisms of the cluster structure on $${{\,\mathrm{\text{ Gr }}\,}}^\circ (k,n)$$, determining a homomorphism from the extended affine braid group to the cluster modular group for $${{\,\mathrm{\text{ Gr }}\,}}(k,n)$$. We also define a quasi-isomorphism between the Grassmannian $${{\,\mathrm{\text{ Gr }}\,}}(k,rk)$$ and the Fock–Goncharov configuration space of 2r-tuples of affine flags for $${{\,\mathrm{\text {SL}}\,}}_k$$. This identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures. Fomin and Pylyavskyy proposed a description of the cluster combinatorics for $${{\,\mathrm{\text{ Gr }}\,}}(3,n)$$ in terms of Kuperberg’s basis of non-elliptic webs. As our main application, we prove many of their conjectures for $${{\,\mathrm{\text{ Gr }}\,}}(3,9)$$ and give a presentation for its cluster modular group. We establish similar results for $${{\,\mathrm{\text{ Gr }}\,}}(4,8)$$. These results rely on the fact that both of these Grassmannians have finite mutation type.

Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

AbstractIn this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.

Regular automorphisms of order p2

Let G be a group, and let α be a regular automorphism of order p2 of G, where p is a prime. If G is polycyclic-by-finite and the map φ: G → G defined by gφ = [g, α] is surjective, then G is soluble. If G is polycyclic, then CG(αp) and G/[G, αp] are both nilpotent-by-finite.

Commensurating actions of birational groups and groups of pseudo-automorphisms

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Property (T), is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.

Some Periodic Groups Admitting a Finite Regular Automorphism of Even Order

We study the structure of an infinite group with automorphism of order 2p, where p is an odd prime leaving only the identity element fixed.

On Automorphisms of a Distance-Regular Graph with Intersection Array {69, 56, 10; 1, 14, 60}

Let Γ be a distance-regular graph of diameter 3 with eigenvalues θ0 > θ1 > θ2 > θ3. If θ2 = −1, then the graph Γ3 is strongly regular and the complementary graph \({\bar \Gamma _3}\) is pseudogeometric for pGc3(k, b1/c2). If Γ3 does not contain triangles and the number of its vertices v is less than 800, then Γ has intersection array {69, 56, 10; 1, 14, 60}. In this case Γ3 is a graph with parameters (392, 46, 0, 6) and \({\bar \Gamma _2}\) is a strongly regular graph with parameters (392, 115, 18, 40). Note that the neighborhood of any vertex in a graph with parameters (392, 115, 18, 40) is a strongly regular graph with parameters (115, 18, 1, 3) and its existence is unknown. In this paper, we find possible automorphisms of these strongly regular graphs and automorphisms of a hypothetical distance-regular graph with intersection array {69, 56, 10; 1, 14, 60}. In particular, it is proved that the latter graph is not ar-ctransitive.