On the weights of the fixed points of an automorphism of a compact Riemann surface

In this work we study automorphisms of compact Riemann surfaces with more than four fixed points. We obtain a lower bound for the weight of each of these fixed points. The discussion depends on the parities of the order of the automorphism and the number of fixed points. Moreover, we discuss the sharpness of our bounds. Received: 15 February 2005     

On orientation-preserving automorphisms of Hamiltonian cycles in the <Emphasis Type="Italic">N</Emphasis>-dimensional Boolean cube

We obtain an upper bound for the order of the group of orientation-preserving automorphisms of a Hamiltonian cycle in the Boolean n-cube. We prove that the existence of a Hamiltonian cycle with the order of the group of orientation-preserving automorphisms attaining this upper bound is equivalent to the existence of a Hamiltonian cycle with an additional condition on the graph of orbits of a fixed automorphism of the n-cube.     

On automorphisms of Enriques surfaces and their entropy

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering K3 surface. We prove a bound of the order of the lift acting on the anti‐invariant cohomology sublattice of the Enriques involution. We use it to obtain some mod 2 constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value. In the Appendix, we give a complete list of Salem numbers that potentially could be the minimal dynamical degree on Enriques surfaces and for which the existence of geometric automorphisms is unknown.     

Simple cycles in the n-cube with a large group of automorphisms

The cycle automorphism in the n-cube is an automorphism of the cube which keeps some cycle in its place and does not change its orientation. An upper bound is found for the order of the group of cycle automorphisms in the n-cube. We obtain the construction for building the long simple cycles for which the order of the group reaches the upper bound.     

Automorphism groups of compact planar Klein surfaces

By a compact Klein surface X, we shall mean a compact surface together with a dianalytic structure on X [1]. A dianalytic homeomorphism of X onto itself will be called an automorphism of X, and we call the automorphism group of X, the full group of automorphisms of X.In this paper we calculate all groups that are the automorphism group of a compact planar Klein surface of algebraic genus p2. As a consequence of the equivalence between compact bordered Klein surfaces and real algebraic curves, we calculate all groups that are the automorphism group of an M-real algebraic curve. Some results on automorphism groups of Riemann surfaces of M-curves were obtained by Natanzon in [14].Partially supported by CAICYT (2280/83)     

Enumeration of the doubles of the projective plane of order 4

A classification of the doubles of the projective plane of order 4 with respect to the order of the automorphism group is presented and it is established that, up to isomorphism, there are 1 746 461 307 doubles. We start with the designs possessing non-trivial automorphisms. Since the designs with automorphisms of odd prime orders have been constructed previously, we are left with the construction of the designs with automorphisms of order 2. Moreover, we establish that a 2-(21,5,2) design cannot be reducible in two inequivalent ways. This makes it possible to calculate the number of designs with only the trivial automorphism, and consequently the number of all double designs. Most of the computer results are obtained by two different approaches and implementations.     

On the automorphism group of a cone

Let V be a real finite dimensional vector space, and let C be a full cone in C. In Sec. 3 we show that the group of automorphisms of a compact convex subset of V is compact in the uniform topology, and relate the group of automorphisms of C to the group of automorphisms of a compact convex cross-section of C. This section concludes with an application which generalizes the result that a proper Lorentz transformation has an eigenvector in the light cone. In Sec. 4 we relate the automorphism group of C to that of its irreducible components. In Sec. 5 we show that every compact group of automorphisms of C leaves a compact convex cross-section invariant. This result is applied to show that if C is a full polyhedral cone, then the automorphism group of C is the semidirect product of the (finite) automorphism group of a polytopal cross-section and a vector group whose dimension is equal to the number of irreducible components of C. An example shows that no such result holds for more general cones.     

Factor group of the automorphism group of a matroid basis graph with respect to the automorphism group of the matroid

Any automorphism of a matroid induces an automorphism of its basis graph. We try to determine what can be said concerning the automorphisms of the basis graph which are not induced by matroids' automorphisms. In particular, we determine the structure of the factor group of the automorphism group of the basis graph with respect to the automorphism group of the matroid, in the event that this factor group exists.     

The structure of skew products with ergodic group automorphisms

We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.     

Some families of polynomial automorphisms

We give a family of polynomial automorphisms of the complex affine plane whose generic length is 3 and degenerating in an automorphism of length 1 with surprisingly high degree.     

Automorphisms of groups of orderp 5

We study automorphisms of groups of orderp ⁵ (p is an odd prime number). Groups without any automorphism of order 2 and groups with group automorphisms of orderp ⁶ are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 562–565, April, 1995.     

Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.     

Polydifferentials and the deformation functor of curves with automorphisms

We give a relation between the dimension of the tangent space of the deformation functor of curves with automorphisms and the Galois module structure of the space of 2-holomorphic differentials. We prove a homological version of the local-global principle similar to the one of J. Bertin and A. Mézard. Let G be a cyclic subgroup of the group of automorphisms of a curve X, so that the order of G is equal to the characteristic. By using the results of S. Nakajima on the Galois module structure of the space of 2-holomorphic differentials, we compute the dimension of the tangent space of the deformation functor.     

On conjugacy in groups of finite-state automorphisms of rooted trees

We show that the conjugacy of elements of finite order in the group of finite-state automorphisms of a rooted tree is equivalent to their conjugacy in the group of all automorphisms of the rooted tree. We establish a criterion for conjugacy between a finite-state automorphism and the adding machine in the group of finite-state automorphisms of a rooted tree of valency 2. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1357–1366, October, 2008.     

Automorphism groups of real algebraic curves which are double covers of the real projective plane

For each integer g≥ 3 we give the complete list of groups acting as full automorphism groups of real algebraic curves of genus $g$ which are double covers of the real projective plane. Explicit polynomial equations of such curves and the formulae of their automorphisms are also given. Received: 29 April 1999 / Revised version: 26 November 1999     

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M _n ) of a free metabelian Lie algebra M _n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M ₄) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M ₃) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.