Continuous automorphisms of Cremona groups

We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.

Strong map-symmetry of SL(3,K) and PSL(3,K) for every finite field K

In this paper, we show that for any finite field [Formula: see text], any pair of map-generators (that is when one of the generators is an involution) of [Formula: see text] and [Formula: see text] has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group [Formula: see text] or [Formula: see text] is reflexible, or equivalently, there are no chiral regular maps with automorphism group [Formula: see text] or [Formula: see text]. As remarked by Leemans and Liebeck, also [Formula: see text] and [Formula: see text] are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

Some cells in the weighted Coxeter group (C̃n,ℓ̃2n)

The affine Coxeter group ([Formula: see text]) can be realized as the fixed point set of the affine Coxeter group ([Formula: see text]) under a certain group automorphism [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the length function of [Formula: see text]. Then the left and two-sided cells of the weighted Coxeter group ([Formula: see text]) can be described explicitly as subsets of ([Formula: see text]). We study the cells of ([Formula: see text]) in the set [Formula: see text] with [Formula: see text] for any [Formula: see text] with [Formula: see text]. Our main result is to show that [Formula: see text] is a two-sided cell of [Formula: see text] which is two-sided connected and that any left cell of [Formula: see text] in [Formula: see text] is left-connected.

On the Isomorphism Problem for Cayley Graphs of Abelian Groups whose Sylow Subgroups are Elementary Abelian or Cyclic

We show that if certain arithmetic conditions hold, then the Cayley isomorphism problem for abelian groups, all of whose Sylow subgroups are elementary abelian or cyclic, reduces to the Cayley isomorphism problem for its Sylow subgroups. This yields a large number of results concerning the Cayley isomorphism problem, perhaps the most interesting of which is the following: if $p_1,\ldots, p_r$ are distinct primes satisfying certain arithmetic conditions, then two Cayley digraphs of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$, $a_i\le 5$, are isomorphic if and only if they are isomorphic by a group automorphism of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$. That is, that such groups are CI-groups with respect to digraphs.

Some New Groups which are not CI-groups with Respect to Graphs

A group $G$ is a CI-group with respect to graphs if two Cayley graphs of $G$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$. We show that an infinite family of groups which include $D_n\times F_{3p}$ are not CI-groups with respect to graphs, where $p$ is prime, $n\not = 10$ is relatively prime to $3p$, $D_n$ is the dihedral group of order $n$, and $F_{3p}$ is the nonabelian group of order $3p$.

Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART I: GENERAL THEORY

Let $G$ be a totally disconnected, locally compact group. A closed subgroup of $G$ is locally normal if its normalizer is open in $G$. We begin an investigation of the structure of the family of closed locally normal subgroups of $G$. Modulo commensurability, this family forms a modular lattice ${\mathcal{L}}{\mathcal{N}}(G)$, called the structure lattice of $G$. We show that $G$ admits a canonical maximal quotient $H$ for which the quasicentre and the abelian locally normal subgroups are trivial. In this situation ${\mathcal{L}}{\mathcal{N}}(H)$ has a canonical subset called the centralizer lattice, forming a Boolean algebra whose elements correspond to centralizers of locally normal subgroups. If $H$ is second-countable and acts faithfully on its centralizer lattice, we show that the topology of $H$ is determined by its algebraic structure (and thus invariant by every abstract group automorphism), and also that the action on the Stone space of the centralizer lattice is universal for a class of actions on profinite spaces. Most of the material is developed in the more general framework of Hecke pairs.

On properties of commutative Alexander quandles

An Alexander quandle Mt is an abelian group M with a quandle operation a * b = ta + (1 - t)b where t is a group automorphism of the abelian group M. In this paper, we will study the commutativity of an Alexander quandle and introduce the relationship between Alexander quandles Mt and M1-t determined by group automorphisms t and 1 - t, respectively.

Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

Given a group automorphismϕ: Γ → Γ, one has an action of Γ on itself byϕ-twisted conjugacy, namely,g.x = gxϕ(g-1). The orbits of this action are calledϕ-twisted conjugacy classes. One says that Γ has theR∞-property if there are infinitely manyϕ-twisted conjugacy classes for every automorphismϕof Γ. In this paper we show that SL(n; Z) and its congruence subgroups have the R8-property. Further we show that any (countable) abelian extension of Γ has the R8-property where Γ is a torsion free non-elementary hyperbolic group, or SL(n; Z); Sp(2n; Z) or a principal congruence subgroup of SL(n; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.