Daniel Franz | Searcharxiv

arXiv2021

Synthesis of Multifunctional Charge Transfer Agents: Towards Single Walled Carbon Nanotubes with Defined Covalent Functionality and Preserved {\pi} System

The attachment of well-defined charge transfer agents to the surface of nanomaterials is an efficient strategy to control their charge density and also to tune their optical, electrical, and physicochemical properties. Particularly interesting are charge transfer agents that either donate or withdraw electrons depending on the arrangements of their building units and that promise a non-destructive attachment to delicate nanomaterials like sp$^2$ compounds. In this work, we rationally synthesize molecular moieties with versatile functionalities. A reactive anchor group allows to attach them to carbon nanotubes as defined charge transfer agents while preserving the tube $\pi$-conjugation. The charge transfer agents were synthesized through the stepwise nucleophilic substitution of either one (monosubstituted series) or two chlorine (disubstituted series) atoms of cyanuric chloride by aniline derivatives containing one, two or three methoxy groups in the para and meta positions. Variation in the number and position of methoxy as an electron transferring group help us to manipulate the electronic and optical properties of the molecular probes and their charge transfer to the SWNTs systematically. The correlation between the optical properties of these molecular probes and their functionality was investigated by experiments and quantum chemical calculations. While the optoelectronic properties of the conjugated charge transfer agents were dominated by the aniline segments, the triazine warrants the ability to nondestructively attach to the surface of SWNTs. This study is one step ahead towards the production of SWNTs with desired optical and electrical properties by covalent $\pi$-preserving functionalization.

cond-mat.mes-hall↗

arXiv2020

Effective finite generation for [IA_n,IA_n] and the Johnson kernel

Let $IA_n$ denote the group of $IA$-automorphisms of a free group of rank $n$, and let $\mathcal I_n^b$ denote the Torelli subgroup of the mapping class group of an orientable surface of genus $n$ with $b$ boundary components, $b=0,1$. In 1935 Magnus proved that $IA_n$ is finitely generated for all $n$, and in 1983 Johnson proved that $\mathcal I_n^b$ is finitely generated for $n\geq 3$. It was recently shown that for each $k\in\mathbb N$, the $k^{\rm th}$ terms of the lower central series $\gamma_k IA_n$ and $\gamma_k\mathcal I_n^b$ are finitely generated when $n>>k$; however, no information about finite generating sets was known for $k>1$. The main goal of this paper is to construct an explicit finite generating set for $\gamma_2 IA_n = [IA_n,IA_n]$ and almost explicit finite generating sets for $\gamma_2\mathcal I_n^b$ and the Johnson kernel, which contains $\gamma_2\mathcal I_n^b$ as a finite index subgroup.

math.GR↗

arXiv2016

Quantifying Residual Finiteness of Linear Groups

Normal residual finiteness growth measures how well a finitely generated group is approximated by its finite quotients. We show that any linear group $\Gamma \leq \mathrm{GL}_d(K)$ has normal residual finiteness growth asymptotically bounded above by $(n\log n)^{d^2-1}$; notably this bound depends only on the degree of linearity of $\Gamma$. We also give precise asymptotics in the case that $\Gamma$ is a subgroup of a higher rank Chevalley group $G$ and compute the non-normal residual finiteness growth in these cases. In particular, finite index subgroups of $G(\mathbb{Z})$ and $G(\mathbb{F}_p[t])$ have normal residual finiteness growth $n^{\dim(G)}.$

math.GR↗