arXiv subjects

Noah Caplinger

Publications and source records attributed to Noah Caplinger.

7 recordsLinked to original sources

arXiv2026

Horocyclic products have Y-posets of hyperbolic structures

A hyperbolic structure on a group $G$ is a (not necessarily properly discontinuous) cobounded action of $G$ on a Gromov hyperbolic space, considered up to coarsely $G$-equivariant quasi-isometry. We show that for groups $G$ acting geometrically and positively on a horocyclic product $X\bowtie Y$, all hyperbolic structures on $G$ come from the two actions on the factors. The ingredients of the proof include Malcev rigidity and a new ("vertical") boundary of horocyclic products. We also give the first example of a group acting geometrically on a horocyclic product of two mixed millefeuille spaces.

math.GR↗

arXiv2025

Groups acting on horocyclic products

Horocyclic products are a well-studied class of metric spaces that provide models for various solvable Lie groups, Baumslag-Solitar groups, and Lamplighter groups. Let $G$ act geometrically on a horocyclic product $X \bowtie Y$ of $\CAT(-κ)$ spaces $X,Y$. We show that every such group is either an ascending HNN extension of a finitely-generated virtually nilpotent group, or else is not finitely presented, depending on the connectivity of the visual boundary of $X\bowtie Y$.

math.GR↗

arXiv2024

Solvable Baumslag-Solitar Lattices

The solvable Baumslag Solitar groups $\text{BS}(1,n)$ each admit a canonical model space, $X_n$. We give a complete classification of lattices in $G_n = \text{Isom}^+(X_n)$ and find that such lattices fail to be strongly rigid$\unicode{x2014}$there are automorphisms of lattices $Γ\subset G_n$ which do not extend to $G_n$$\unicode{x2014}$but do satisfy a weaker form of rigidity: for all isomorphic lattices $Γ_1,Γ_2\subset G_n$, there is an automorphism $ρ\in \text{Aut}(G_n)$ so that $ρ(Γ_1) = Γ_2$.

math.GR↗

arXiv2024

Totally Symmetric Sets

We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.

math.GR↗

arXiv2022

Large totally symmetric sets

A totally symmetric set is a subset of a group such that every permutation of the subset can be realized by conjugation in the group. The (non-)existence of large totally symmetric sets obstruct homomorphisms, so bounds on the sizes of totally symmetric sets are of particular use. In this paper, we prove that if a group has a totally symmetric set of size $k$, it must have order at least $(k+1)!$. We also show that with three exceptions, $\{(1 \; i)\mid i = 2,\ldots,n\} \subset S_n$ is the only totally symmetric set making this bound sharp; it is thus the largest totally symmetric set relative to the size of the ambient group.

math.GR↗

arXiv2022

Totally symmetric sets in the general linear group

A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the algebraic structure. In this paper we introduce a more general perspective on total symmetry, and formulate a notion of "irreducibility" for totally symmetric sets in the general linear group. We classify irreducible totally symmetric sets, as well as those of maximal cardinality.

math.GR↗

arXiv2020

Small Quotients of Braid Groups

We prove that the symmetric group $S_n$ is the smallest non-cyclic quotient of the braid group $B_n$ for $n=5,6$ and that the alternating group $A_n$ is the smallest non-trivial quotient of the commutator subgroup $B_n'$ for $n = 5,6,7,8$. We also give an improved lower bound on the order of any non-cyclic quotient of $B_n$.

math.GT↗