arXiv subjects

Samuel M. Corson

Publications and source records attributed to Samuel M. Corson.

34 records · Page 2Linked to original sources

arXiv2020

Bi-orders do not arise from total orders

We present some Zermelo-Fraenkel consistency results regarding bi-orderability of groups, as well as a construction of groups with Conradian orders whose every action on metric spaces has bounded orbits. A classical consequence of the ultrafilter lemma is that a group is bi-orderable if and only if it is locally bi-orderable. We show that there exists a model of ZF in which there is a group which is locally free (ergo locally bi-orderable) and not bi-orderable, and the group can be given a total order. Such a group can also exist in the presence of the principle of dependent choices. Comparable consistency results are provided for torsion-free abelian groups.

math.GR↗

arXiv2020

The independence of Stone's Theorem from the Boolean Prime Ideal Theorem

We give a permutation model in which Stone's Theorem (every metric space is paracompact) is false and the Boolean Prime Ideal Theorem (every ideal in a Boolean algebra extends to a prime ideal) is true. The erring metric space in our model attains only rational distances and is not metacompact. Transfer theorems give the comparable independence in the Zermelo-Fraenkel setting, answering a question of Good, Tree and Watson.

math.LO↗

arXiv2020

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

We describe homomorphisms $φ:H\rightarrow G$ for which the codomain is acylindrically hyperbolic and the domain is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of $φ$ is small or $φ$ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to $G$ as above. We prove a more precise result for homomorphisms $φ:H\rightarrow {\rm Mod}(Σ)$, where $H$ as above and ${\rm Mod}(Σ)$ is the mapping class group of a connected compact surface $Σ$. In this case there exists an open normal subgroup $V\leqslant H$ such that $φ(V)$ is finite. We also prove the analogous statement for homomorphisms $φ:H\rightarrow {\rm Out}(G)$, where $G$ is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

math.GR↗

arXiv2019

A Nunke type classification in the locally compact setting

In this short note we prove that a group G is lcH-slender -- that is, every abstract group homomorphism from a locally compact Hausdorff topological group to G has an open kernel -- if and only if G is torsion-free and does not include Q or the p-adic integers Zp for any prime p. This mirrors a classical characterization given by Nunke for slender abelian groups.

math.GR↗

arXiv2019

Strongly bounded groups of various cardinalities

Strongly bounded groups are those groups for which every action by isometries on a metric space has orbits of finite diameter. Many groups have been shown to have this property, and all the known infinite examples so far have cardinality at least $2^{\aleph_0}$. We produce examples of strongly bounded groups of many cardinalities, including $\aleph_1$, answering a question of Yves de Cornulier [4]. In fact, any infinite group embeds as a subgroup of a strongly bounded group which is, at most, two cardinalities larger.

math.GR↗

arXiv2019

Freely indecomposable almost free groups with free abelianization

For certain uncountable cardinals $κ$ we produce a group of cardinality $κ$ which is freely indecomposable, strongly $κ$-free, and whose abelianization is free abelian of rank $κ$. The construction takes place in Gödel's constructible universe $L$. This strengthens an earlier result of Eklof and Mekler.

math.GR↗

arXiv2019

On preservation of automatic continuity

A group $G$ is called automatically continuous if any homomorphism from a completely metrizable or locally compact Hausdorff group to $G$ has open kernel. In this paper, we study preservation of automatic continuity under group-theoretic constructions, focusing mainly on groups of size less than continuum. In particular, we consider group extensions and graph products. As a consequence, we establish automatic continuity of virtually poly-free groups, and hence of non-exceptional spherical Artin groups. On the other hand, we show that if $G$ is automatically continuous, then so is any finitely generated residually $G$ group, hence, for instance, all finitely generated residually free groups are automatically continuous.

math.GR↗

arXiv2018

Automatic continuity of $\aleph_1$-free groups

We prove that groups for which every countable subgroup is free ($\aleph_1$-free groups) are n-slender, cm-slender, and lcH-slender. In particular every homomorphism from a completely metrizable group to an $\aleph_1$-free group has an open kernel. We also show that $\aleph_1$-free abelian groups are lcH-slender, which is especially interesting in light of the fact that some $\aleph_1$-free abelian groups are neither n- nor cm-slender. The strongly $\aleph_1$-free abelian groups are shown to be n-, cm-, and lcH-slender. We also give a characterization of cm- and lcH-slender abelian groups.

math.GR↗

arXiv2018

Deeply concatenable subgroups might never be free

We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer-Specker group $\mathbb{Z}^ω$ and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented

math.GR↗

arXiv2018

The number of homomorphisms from the Hawaiian earring group

We show a dichotomy for groups of cardinality less than continuum. The number of homomorphisms from the Hawaiian earring group to such a group $G$ is either the cardinality of $G$ in case $G$ is noncommutatively slender, or the number is $2^{2^{\aleph_0}}$ in case $G$ is not noncommutatively slender. An example of a noncommutatively slender group with nontrivial divisible element is exhibited.

math.GR↗

arXiv2017

A note on automatic continuity

We present new results regarding automatic continuity, unifying some diagonalization concepts that have been developed over the years. For example, any homomorphism from a completely metrizable topological group to Thompson's group $F$ has open kernel. A similar claim holds when $F$ is replaced with a Baumslag-Solitar group or a torsion-free word hyperbolic group.

math.GR↗

arXiv2017

On definable subgroups of the fundamental group

We present several new theorems concerning the first fundamental group of a path connected metric space. Among the results proven are strengthenings of the main theorems of \cite{Sh2} and \cite{CoCo}. A compactness theorem for the fundamental group of a Peano continuum is given. A useful characterization for the shape kernel of a locally path connected space is presented, along with a very succinct proof of the fact that for such a space the Spanier and shape kernel subgroups coincide (see \cite{BF}). We also show that a free decomposition of the fundamental group of a locally path connected Polish space cannot be nonconstructive. Numerous other results and examples illustrating the sharpness of our theorems are provided.

math.GN↗

arXiv2016

On subgroups of first homology

We prove several new theorems regarding first homology. Some dichotomies for first homology of Peano continua are presented, as well as a notion of strong abelianization for arbitrary path connected metric spaces. We also show that the fundamental group of the Hawaiian earring has Borel subgroups of almost all multiplicative and additive types.

math.AT↗

arXiv2016

On quasi-isometry and choice

In this note we prove that the symmetry of the quasi-isometry relation implies the axiom of choice, even when the relation is restricted to geodesic hyperbolic spaces. We show that this result is sharp by demonstrating that symmetry of quasi-isometry in an even more restrictive setting does not imply the axiom of choice. The "Bottleneck Theorem" of Jason Fox Manning also implies choice.

math.LO↗

arXiv2015

Torsion-free Word Hyperbolic Groups are Noncommutatively Slender

In this note we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Graham Higman, in his seminal 1952 paper, proved that free groups are noncommutatively slender. Such groups were first defined by K. Eda. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few- relator sense of Gromov are noncommutatively slender.

math.GR↗

arXiv2015

On the First Homology of Peano Continua

We show that the first homology group of a locally connected compact metric space is either uncountable or is finitely generated. This is related to Shelah's well-known result which shows that the fundamental group of such a space satisfies a similar criterion. We give an example of such a space whose fundamental group is uncountable but whose first homology is trivial, showing that our result doesn't follow from Shelah's. We clarify a claim made by Pawlikowski and offer a proof of the clarification.

math.GN↗