Digraphs with small automorphism groups that are Cayley on two nonisomorphic groups

Let $\Gamma=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(\Gamma)$. The Cayley index of $\Gamma$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $\Gamma$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $\Gamma$ on $G$ whose Cayley index is just $p$, and whose automorphism group contains a nonabelian regular subgroup

Groups for which it is easy to detect graphical regular representations

We say that a finite group G is "DRR-detecting" if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.Comment: 11 pages. v2: added acknowledgments and author addres

M\mathcal{M}-coextensivity and the strict refinement property

The notion of an $\mathcal{M}$-coextensive object is introduced in an arbitrary category $\mathbb{C}$, where $\mathcal{M}$ is a distinguished class of morphisms from $\mathbb{C}$. This notion allows for a categorical treatment of the strict refinement property in universal algebra, and highlights its connection with extensivity in the sense of Carboni, Lack and Walters. If $\mathcal{M}$ is the class of all product projections in a variety of algebras $\mathbb{C}$, then the $\mathcal{M}$-coextensive (or projection-coextensive) objects in $\mathbb{C}$ turn out to be precisely those algebras which have the strict refinement property. If $\mathcal{M}$ is the class of surjective homomorphisms in the variety, then the $\mathcal{M}$-coextensive objects are precisely those algebras which have directly-decomposable (or factorable) congruences. In exact Mal'tsev categories, every centerless object with global support has the strict refinement property. We will also show that in every exact majority category, every object with global support has the strict refinement property.Comment: 26 page

The semigroup of metric measure spaces and its infinitely divisible probability measures

A metric measure space is a complete separable metric space equipped with probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov-Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation $\boxplus$ on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric and that each element has a unique factorization into prime elements. We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. For example, we show that for any given positive real numbers $a,b,c$ the trivial space is the only space $\mathcal{X}$ that satisfies $a \mathcal{X} \boxplus b \mathcal{X} = c \mathcal{X}$. We establish that there is no analogue of the law of large numbers: if $\mathbf{X}_1, \mathbf{X}_2$..., is an identically distributed independent sequence of random spaces, then no subsequence of $\frac{1}{n} \boxplus_{k=1}^n \mathbf{X}_k$ converges in distribution unless each $\mathbf{X}_k$ is almost surely equal to the trivial space. We characterize the infinitely divisible probability measures and the L\'evy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class.Comment: 48 pages, 0 figures. The previous version considered only compact metric measure spaces, but new arguments allow all the results of the previous version to be extended to the setting of complete separable metric measure space