The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Can one hear the shape of a drum

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

3-Manifold Groups

We strengthen the analogy between convex cocompact Kleinian groups and convex cocompact subgroups of the mapping class group of a surface in the sense of B. Farb and L. Mosher.

Shadows Of Mapping Class Groups: Capturing Convex Cocompactness

From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these

/pdf/on-groups-generated-by-two-positive-multi-twists-teichmuller-34z0n5h4og.pdf

On groups generated by two positive multi-twists: Teichmuller curves and Lehmer's number

In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we give an embedding into the space of geodesic currents and use this to obtain a compactification for the space of flat metrics. The geometric interpretation is that flat metrics degenerate to mixed structures on the surface: part flat metric and part measured foliation.

/pdf/length-spectra-and-degeneration-of-flat-metrics-1kssadipq5.pdf

Length spectra and degeneration of flat metrics

We describe sucient conditions which guarantee that a nite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi- isometric embedding for both of the standard metrics. As a consequence, we produce innitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmuller space.

/pdf/the-geometry-of-right-angled-artin-subgroups-of-mapping-1e2po3s9y9.pdf

The geometry of right angled Artin subgroups of mapping class groups

The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface $S_g$ of genus $g$. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of $S_g$ tends to zero at the rate $1/g$. We consider here the smallest dilatation of any pseudo-Anosov homeomorphism of $S_g$ acting trivially on $\Gamma/\Gamma_k$, the quotient of $\Gamma = \pi_1(S_g)$ by the $k^{\rm th}$ term of its lower central series, $k \geq 1$. In contrast to Penner's asymptotics, we prove that this minimal dilatation is bounded above and below, independently of $g$, with bounds tending to infinity with $k$. For example, in the case of the Torelli group ${\cal I}(S_g)$, we prove that $L({\cal I}(S_g))$, the logarithm of the minimal dilatation in ${\cal I}(S_g)$, satisfies $.197 < L({\cal I}(S_g))< 4.127$. In contrast, we find pseudo-Anosov mapping classes acting trivially on $\Gamma/\Gamma_k$ whose asymptotic translation lengths on the complex of curves tend to $0$ as $g\to\infty$.

https://www.jstor.org/stable/info/40068147

Christopher J. Leininger

Papers

Shadows Of Mapping Class Groups: Capturing Convex Cocompactness

On groups generated by two positive multi-twists: Teichmuller curves and Lehmer's number

Length spectra and degeneration of flat metrics

The geometry of right angled Artin subgroups of mapping class groups

The lower central series and pseudo-Anosov dilatations