arXiv subjects
Publications and source records attributed to Samuel M. Corson.
It is shown that if a metric space exhibits certain finiteness and tree-like properties, then its group of bounded displacement does not include a subgroup isomorphic to the dyadic rational numbers. This extends a result of N. M. Suchkov, A. A. Shlepkin, and D. A. Taysnyov.
We prove a general theorem giving constraints on maps from certain topological groups to inverse limits of bounded torsion groups. From this we obtain some automatic continuity and ultraproduct results. For example, every homomorphism from a Polish group to a countable torsion-free residually finite group has open kernel. Also, the Grigorchuk group is a homomorphic image of a nonprincipal ultraproduct of groups if and only if there exists a measurable cardinal.
We prove that every right-angled Coxeter group (RACG) is profinitely rigid amongst all Coxeter groups. On the other hand we exhibit RACGs which have infinite profinite genus amongst all finitely generated residually finite groups. We also establish profinite rigidity results for graph products of finite groups. Along the way we prove that the Higman-Thompson groups $V_{n}$ are generated by $4$ involutions, generalising a classical result of Higman for Thompson's group $V$.
We prove the conjecture of James W. Cannon and Gregory R. Conner that the fundamental group of the Griffiths double cone space is isomorphic to that of the harmonic archipelago. From this and earlier work in this area, we conclude that the isomorphism class of these groups is quite large and includes groups with a great variety of descriptions.
Utilizing an embedding theorem of Obraztsov we construct groups as described in the title. This provides an affirmative answer to a problem of D. O. Revin. The constructed groups also provide a negative answer to a question highlighted by Kurdachenko, Russo, and Vincenzi.
In this short note we answer some questions of Bergman regarding homomorphic images of (ultra)products of groups.
It is shown, from $σ$-centered Martin's Axiom, that there exists a proper dense subgroup of the symmetric group on a countably infinite set whose natural action on sufficiently flexible relational structures is transitive. This allows us to give consistent positive answers to some questions of Peter M. Neumann from the 1980s.
A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We show that this problem is essentially the same as an analogous question, regarding universal algebras, asked by Jónsson in the 1960s. We further show that these problems are the same as the so-called free subset problem. As a result, one can have a consistent strong negative answer (from a large cardinal assumption) as well as a consistent positive answer.
We prove that affine Coxeter groups are profinitely rigid.
It is shown that if $\{H_n\}_{n \in ω}$ is a sequence of groups without involutions, with $1 < |H_n| \leq 2^{\aleph_0}$, then the topologist's product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in ω}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in ω} A_n/\bigoplus_{n \in ω} A_n$ is dependent on the sequence.
We produce a simple group $G$ of cardinality $\aleph_1$ which is Artinian (every strictly descending chain of subgroups is finite), satisfies a Burnside law and such that for each uncountable subset $Y \subseteq G$ there exists a natural number $n_Y$ for which every element of $G$ may be expressed as a product of length at most $n_Y$ of elements in $Y^{\pm 1}$. In particular this group is Jónsson (every proper subgroup is of strictly smaller cardinality) and strongly bounded (every abstract action on a metric space has bounded orbits); this is the first example of an uncountable group having both of these properties which is constructed without using the continuum hypothesis. The group $G$ can also be made so that all subgroups are simple and all nontrivial subgroups are malnormal in $G$.
We classify pointed spaces according to the first fundamental group of their reduced suspension. A pointed space is either of so-called totally path disconnected type or of horseshoe type. These two camps are defined topologically but a characterization is given in terms of fundamental groups. Among totally path disconnected spaces the fundamental group is shown to be a complete invariant for a notion of topological equivalence weaker than that of homeomorphism.
Let $κ$ be an inaccessible cardinal, $\mathfrak{U}$ be a universal algebra, and $\sim$ be the equivalence relation on $\mathfrak{U}^κ$ of eventual equality. From mild assumptions on $κ$ we give general constructions of $\mathcal{E} \in End(\mathfrak{U}^κ/\sim)$ satisfying $\mathcal{E} \circ \mathcal{E} = \mathcal{E}$ which do not descend from $Δ\in End(\mathfrak{U}^κ)$ having small strong supports. As an application there exists an $\mathcal{E} \in End(\mathbb{Z}^κ/\sim)$ which does not come from a $Δ\in End(\mathbb{Z}^κ)$.
This very short correction notes a gap in an argument of an earlier paper, and also provides a theorem of similar flavor to the main result of that paper.
A group $G$ is Jónsson if $|H| < |G|$ whenever $H$ is a proper subgroup of $G$. Using an embedding theorem of Obraztsov it is shown that there exists a Jónsson group $G$ of infinite cardinality $κ$ if and only if there exists a Jónsson algebra of cardinality $κ$. Thus the question as to which cardinals admit a Jónsson group is wholly reduced to the well-studied question of which cardinals are not Jónsson. As a consequence there exist Jónsson groups of arbitrarily large cardinality. Another consequence is that the infinitary edge-orbit conjecture of Babai is true.
It is shown that the fundamental group of the Griffiths double cone space is isomorphic to that of the triple cone. More generally if $κ$ is a cardinal such that $2 \leq κ\leq 2^{\aleph_0}$ then the $κ$-fold cone has the same fundamental group as the double cone. The isomorphisms produced are non-constructive, and no isomorphism between the fundamental group of the $2$- and of the $κ$-fold cones, with $2 < κ$, can be realized via continuous mappings. We also prove a conjecture of James W. Cannon and Gregory R. Conner which states that the fundamental group of the Griffiths double cone space is isomorphic to that of the harmonic archipelago.
We give the construction of an infinite topological space with unusual properties. The space is regular, separable, and connected, but removing any nonempty open set leaves the remainder of the space totally disconnected (in fact, totally separated). The space is also strongly Choquet (in fact, satisfies an even stronger condition) and has a basis with nice properties. The construction utilizes Jensen's diamond principle $\diamondsuit$.
We introduce the notion of limiting theories, giving examples and providing a sufficient condition under which the first order theory of a structure is the limit of the first order theories of a collection of substructures. We also give a new proof that theories like that of infinite sets are not finitely axiomatizable.