arXiv subjects

Sang Rae Lee

Publications and source records attributed to Sang Rae Lee.

4 recordsLinked to original sources

arXiv2015

Growth rate of endomorphisms of Houghton's groups

A Houghton's group $\mathcal{H}_n$ consists of translations at infinity of a $n$ rays of discrete points on the plane. In this paper we study the growth rate of endomorphisms of Houghton's groups. We show that if the kernel of an endomorphism $ϕ$ is not trivial then the growth rate $\mathrm{GR}(ϕ)$ equals either $1$ or the spectral radius of the induced map on the abelianization. It turns out that every monomorphism $ϕ$ of $\mathcal{H}_n$ determines a unique natural number $\ell$ such that $ϕ(\mathcal{H}_n)$ is generated by translations with the same translation length $\ell$. We use this to show that $\mathrm{GR}(ϕ)$ of a monomorphism $ϕ$ of $\mathcal{H}_n$ is precisely $\ell$ for all $2\leq n$.

math.GR↗

arXiv2015

The $R_{\infty}$ property for Houghton's groups

We study twisted conjugacy classes of a family of groups which are called Houghton's groups $\mathcal{H}_n$ ($n \in\mathbb{N}$), the group of translations of $n$ rays of discrete points at infinity. We prove that the Houghton's groups $\mathcal{H}_n$ have the $R_{\infty}$ property for all $n\in \mathbb{N}$.

math.GR↗

arXiv2012

Geometry of Houghton's Groups

Hougthon's groups H_n is a family of groups where each H_n consists of `translations at infinity' on n rays of discrete points emanating from the origin on the plane. Brown shows H_n has type FP_n-1 but not FP_n by constructing infinite dimensional cell complex on which H_n acts with certain conditions. We modify his idea to construct n-dimensional CAT(0) cubical complex X_n on which H_n acts with the same conditions as before. Brown also shows H_n is finitely presented provided n>2. Johnson provides a finite presentation for H_3. We extend his result to provide finite presentations of H_n for n>3. We also establish exponential isoperimetric inequalities of H_n for n>2.

math.GR↗

arXiv2011

Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and have exponential, or polynomial Dehn functions of prescribed degree.

math.GR↗