The local-to-global property for Morse quasi-geodesics

We show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

Predictors of Failure of Return to Play in Youth Baseball Players After Capitellar Osteochondritis Dissecans: Focus on Elbow Valgus Laxity and Radiocapitellar Congruity

Background: Osteochondritis dissecans of the humeral capitellum (capitellar OCD) is a common injury among youth baseball players, but there are only a few studies that report on return to play with nonoperative treatment. Purpose: To evaluate the medial elbow joint laxity under valgus stress and radiocapitellar congruity in patients with capitellar OCD and evaluate their relationship to predicting rehabilitation outcome. Study Design: Case-control study; Level of evidence, 3. Methods: Capitellar OCD was diagnosed in 81 patients included in our study. All patients were elementary school students who initially received rehabilitation treatment after injury. The rates of return to the same level of play or higher (RTSP) were calculated and correlated with the joint gap difference between the dominant and nondominant elbows using ultrasound and radiocapitellar congruity (proximal, lateral, and anterior radial translation length), which was assessed using plain radiographs of the dominant elbow. Results: The overall RTSP rates of patients with nonoperative treatment was 70.4% (57/81). The multivariate logistic regression analysis identified OCD classification (stage I, odds ratio [OR], 4.076; 95% CI, 1.171-14.190) and 1 continuous variable (proximal radial translation length on anteroposterior view, OR, 0.661; 95% CI, 0.479-0.911) as the significant predictive factors for outcome after nonoperative treatment. Conclusion: The early stage of capitellar OCD in youth baseball players can be successfully treated nonoperatively in the majority of cases. The presence of proximal radial translation can predict the outcome of nonoperative management of capitellar OCD.

Minimal Asymptotic Translation Lengths of Torelli Groups and Pure Braid Groups on the Curve Graph

In this paper, we show that the minimal asymptotic translation length of the Torelli group ${\mathcal{I}}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group ${\textrm{Mod}}(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translation length of the pure braid group ${\textrm{PB}}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group ${\textrm{B}}_n$, which behaves like $1/n^2$.

Random walks on weakly hyperbolic groups

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the convergence result to show linear progress and linear growth of translation length, without any assumptions on the moments of the random walk. If the action is acylindrical, and the random walk has finite entropy and finite logarithmic moment, we show that the Gromov boundary with the hitting measure is the Poisson boundary.

Statistics and compression of scl

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting non-degenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length$n$is of order$n/ \log n$. This establishes quantitative refinements of qualitative results of Bestvina and Fujiwara and others on the infinite dimensionality of two-dimensional bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense that we can control the geometry of the unit balls in these normed vector spaces (or rather, in random subspaces of their normed duals). As a corollary of our methods, we show that an element obtained by random walk of length$n$in a mapping class group cannot be written as a product of fewer than$O(n/ \log n)$reducible elements, with probability going to$1$as$n$goes to infinity. We also show that the translation length on the complex of free factors of a random walk of length$n$on the outer automorphism group of a free group grows linearly in$n$.

NON-RIGIDITY OF CYCLIC AUTOMORPHIC ORBITS IN FREE GROUPS

We say a subset Σ ⊆ FN of the free group of rank N is spectrally rigid if whenever T1, T2 ∈ cv N are ℝ-trees in (unprojectivized) outer space for which ‖σ‖T1 = ‖σ‖T2 for every σ ∈ Σ, then T1 = T2 in cv N. The general theory of (non-abelian) actions of groups on ℝ-trees establishes that T ∈ cv N is uniquely determined by its translation length function ‖⋅‖T : FN → ℝ, and consequently that FN itself is spectrally rigid. Results of Smillie and Vogtmann, and of Cohen, Lustig and Steiner establish that no finite Σ is spectrally rigid. Capitalizing on their constructions, we prove that for any Φ ∈ Aut (FN) and g ∈ FN, the set Σ = {Φn(g)}n∈ℤ is not spectrally rigid.

GENERICITY OF FILLING ELEMENTS

An element of a finitely generated non-Abelian free group F(X) is said to be filling if that element has positive translation length in every very small minimal isometric action of F(X) on an ℝ-tree. We give a proof that the set of filling elements of F(X) is exponentially F(X)-generic in the sense of Arzhantseva and Ol'shanskiı. We also provide an algebraic sufficient condition for an element to be filling and show that there exists an exponentially F(X)-generic subset consisting only of filling elements and whose membership problem has linear time complexity.

A Calculation Model for the Film Thickness and Component Uniformity of Twin Co-Sputtering Target System

In the vacuum film coating field, the co-sputtering method by two or more targets has been widely used for coating the composite film consisted of a variety of component elements. In this paper, a typical co-sputtering system is studied which composes the twin round planar magnetron sputtering targets settled symmetrically and slantways towards a single flat substrate with self-rotation. A model is set up to describe the non-dimensional relationship between the film thickness and the structural parameters of co-sputtering system, such as distance of the substrate-to-target , symmetrical eccentricity , translation length and obliquity angle of target . On the assumption that the sputtered particles are emitted in the direction of cosine distribution and fly straightly without collision scattering, the depositing rate distribution, film thickness distribution, the utilization ratio of the target sputtering material, and the fluctuant ratio of the components from two targets are calculated. Some of simulating examples are given by use of Matlab software.