Stefano Francaviglia
We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping bounded the length of the uniform current is compact (up to conjugation.) This implies that the spectrum of the length of the images of the uniform current is discrete, answering to a conjecture of I. Kapovich.
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We study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth $ϕ$, under the action of the centraliser $C(ϕ)$. As a corollary, we get that the same holds for the action of $<ϕ>$ on $Min(ϕ)$. Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one is consisted of a single point.
We prove a purely topological characterization of the Moufang property for disconnected compact polygons in terms of convergence groups. As a consequence, we recover the fact that a locally finite thick affine building of rank 3 is a Bruhat-Tits building if and only if its automorphism group is strongly transitive. We also study automorphism groups of general compact polygons without any homogeneity assumption. A compactness criterion for sets of automorphisms is established, generalizing the theorem by Burns and Spatzier that the full automorphism group, endowed with the compact-open topology…
We study the Lipschitz metric on Outer Space and prove that fully irreducible elements of Out(F_n) act by hyperbolic isometries with axes which are strongly contracting. As a corollary, we prove that the axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are stable, meaning that a quasi-geodesic with endpoints on the axis stays within a bounded distance from the axis.
We prove that the group of outer automorphisms of the free Coxeter group $W_n$ is acylindrically hyperbolic in the sense of Osin. As an application, we observe that any CAT(0) space admitting a geometric action by Out($W_n$) must contain a rank-one geodesic. The theorem proceeds from expanding on a well-known relationship between Out($W_n$) and the outer automorphism group of free groups.
North-South dynamics of hyperbolic free group automorphisms on the space of currents
本文证明在正反方向均容许绝对 train track 代表的双曲自由群外自同构，在射影化测地流空间上具有广义一致南北动力学。
Given a tree $T$ and a group $\Ga$ of automorphisms of $T$, we study the markovian properties of the geodesic flow on the quotient by $\Ga$ of the space of geodesics of $T$. For instance, when $T$ is the Bruhat-Tits tree of a semi-simple connected algebraic group $\underline{G}$ of rank one over a non archimedian local field $\wh K$, and $\Ga$ is a (possibly non uniform) lattice in $\underline{G}(\wh K)$, we prove that the type preserving geodesic flow is Bernoulli with finite entropy. Under some mild assumptions, we prove that if the quotient geodesic flow is mixing for a probability Patterso…
Sigma-compactness of Morse boundaries in Morse local-to-global groups and applications to stationary measures
本文证明了 Morse local-to-global 群的 Morse 边界具有 $σ$-紧性，且逆命题对小消去群成立，并以此推导出具有收缩性的非双曲群的 Morse 边界不存在非平凡平稳测度。
In 1981 Masur proved the existence of a dense geodesic in the moduli space for a Teichmüller space. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove that the set of Perron-Frobenius eigenvectors of positive integer $m \times m$ matrices is dense in the positive cone $\mathbb{R}^m_+$ (these matrices will in fact be the transition matrices of positive automorphisms). We give a proof in the appendix that not every point in the b…
Manhattan geodesics and the boundary of the space of metric structures on hyperbolic groups
本文为双曲群度量结构空间构造外自同构不变的测地线双梳及其边界，证明其包含多种伪度量，并由此得到长度谱刚性、正则性、增长率等结果，否定 Bonahon 猜想。
Scissors automorphism groups II: Solomon-Tits theorems
本文证明了在欧几里得、双曲或球面几何中，由点或超平面生成的测地子空间集合的正则非平凡子空间的poset的实现在球面 wedge 中等价，为后续计算剪刀自同构群的同调奠定基础。
Random walks and contracting elements III: Outer space and outer automorphism group
该研究延续此前工作，利用等距映射的BGIP研究非对称度量空间上的随机游走，证明秩≥3的自由群的典型外自同构前向、后向扩张因子不同，解答了Handel与Mosher的问题。
Train Tracks on Graphs of Groups and Outer Automorphisms of Hyperbolic Groups
本研究证明满足条件的群图基本群自同构可用Bestvina-Handel型不可约轨道轨形映射表示，推导自由积自同构范式并证明其非拟测地，还证实有限生成字双曲群外自同构满足推广Nielsen-Thurston分类的动力学三分性。
We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is Euclidean Lipschitz and, consequently, that the metric spheres are boundaries of Lipschitz domains in the Euclidean sense. In the first part of the paper, we consider geodesic distances. In this case, we actually prove the regularity of the distance in the more general context o…
Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications
本文研究连通李群自同构在闭单参数子群紧空间 Sub$^p_G$ 上的 distal 与扩张动力学，证明环面上仅有限阶自同构 distal 作用，并证明中心环面维数至少为 2 的连通李群不存在扩张自同构，推广了若干已有结果。
On the action of relatively irreducible automorphisms on their train tracks
本文研究相对𝒢的指数型𝒢-不可约自同构的轨道轨点集（即最小位移集Min(φ)），证明其在φ生成的循环子群作用下余紧，得到相关辅助结果，并推广了自由群自同构中心化子的经典定理。
Uniform difference between the length spectra of Out(F2) and the genus two handlebody group
本文研究了双曲面二柄体群与 Out(F2) 之间长度谱的均匀差异，揭示了伪 Anosov 映射的伸缩因子与完全不可约外自同构之间的关系。
Using geodesic currents, we provide a theoretical justification for some of the experimental results regarding the behavior of Whitehead's algorithm on non-minimal inputs, that were obtained by Haralick, Miasnikov and Myasnikov via pattern recognition methods. In particular we prove that the images of "random" elements of a free group $F$ under the automorphisms of $F$ form "clusters" that share similar normalized Whitehead graphs and similar behavior with respect to Whitehead's algorithm.
We prove that all atoroidal automorphisms of $Out(F_N)$ act on the space of projectivized geodesic currents with generalized north-south dynamics. As an application, we produce new examples of non virtually cyclic, free and purely atoroidal subgroups of $Out(F_N)$ such that the corresponding free group extension is hyperbolic. Moreover, these subgroups are not necessarily convex cocompact.
Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups
本文刻画了Lie群自同构在单参数子群空间上的远端作用，将其与群上作用及自同构生成子群的紧性关联，并推广至子群作用，证明共轭远端作用等价于群为紧或紧与向量群直积。
A compactness theorem for twisted-unitary elements of integral group rings, with a certified route to the theta-unitary Case A at window B(4) of the Promislow group
本文为整群环的扭曲酉元证明紧致性定理，结合SAT认证与镜像对称，将Promislow群B(4)窗口的theta-酉单位消失问题归约为单个下界，并给出数值估计与结构障碍分析。
On the Hausdorff dimension and attracting laminations for fully irreducible automorphisms of free groups
研究自由群自同构相关抽象代数层叠端点集的豪斯多夫维数，对指数增长及环面完全不可约外自同构，其吸引层叠或终结层叠端点集的豪斯多夫维和填充维数为\(0\)，上盒维数\(>0\)。
Generalisable presentations and compactness properties of locally compact right-angled Artin groups
本文提出可推广表示的概念，引入拓扑右角Artin群，研究其几何与紧性性质，并构造了具有特定有限性性质的离散群例子。
When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor projections to construct an action of Out(F_n) on a finite product of hyperbolic spaces where every automorphism with exponential growth acts with positive translation length. We also prove a version of the Bounded geodesic image theorem. In the appendix, we give a sketch of the p…
Locally elliptic actions, torsion groups, and nonpositively curved spaces
猜想有限生成群在有限维非正曲率复形上的局部椭圆作用具有全局不动点，并以Helly图为主要工具，针对欧几里得建筑、Helly复形等一大类复形证明了该猜想。
Convergence and collapsing of CAT$(0)$-lattices
本文研究CAT$(0)$-格及其商空间的收敛理论，刻画分裂与坍缩现象并解释作用如何退化为非离散极限作用，进而证明紧致CAT$(0)$-同调轨道流形的紧性定理，并给出平坦轨道空间孤立性与熵夹挤定理等应用。
We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time $n$ of the random walk to the Lyapunov exponent of the walk, which gives the typical growth rate of the length of a curve -- or of a conjugacy class in $F_N$ -- under a random product of diffeomorphisms/automorphisms. In the mapping class group case, we first observe that the drift of the random walk in the curve complex is also …
Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat--Tits buildings. We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT(0) space to be Moufang, and deduce that an irreducible…
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