Dynamics and geometry of free-by-cyclic groups

群是一种代数结构，它捕捉几何对象或物理或机械系统的对称性集合的抽象性质。该项目将集中研究自由环基团。这些群是纤维三维流形的基本群的自然对应，纤维三维流形是重要的数学对象，在拓扑学、微分几何、遍历理论、数论、计算复杂性和其他数学分支中发挥着关键作用。与3-流形群相比，被循环自由的群的理解要少得多，尽管它们表现出相当广泛的有趣的特征和行为。该项目旨在研究自由循环群的几何和动力学，特别是通过探索这些群的新的多项式不变量。这项研究有助于更好地理解自由循环群和它们的3-流形“表亲”之间的区别和相似之处，并导致对群论中代数和动力学相互作用的新见解。该项目将建立在提出者与Leininger和Dowdall联合的新工作的基础上，通过与这样一个群相关的“折叠映射环面”来研究按循环自由群的性质。折叠映射环配备了一个自然的连续变换半流，它以一种特别有用和可访问的形式编码了关于群的关键代数、几何和动力学信息。与折叠映射环面相关的另一个关键对象是“麦克马伦多项式”，它是纤维三维流形的TeichMuller多项式的按循环自由模拟。该项目的主要总体研究目标是了解自由循环群G可以分裂为自由循环群和自由群的上升HNN扩张的许多可能方法的代数、几何和动力学特征，并获得关于自由群自同构的新知识。要研究的具体问题包括：理解折叠映射环面的截面锥与群的BNS不变量之间的关系，使稳定树和与G的不同分裂相关的Cannon-瑟斯顿映射一致，研究与G的不同自由循环分裂相关的单调自同构的不可约性，发展G的同调Zeta函数的模拟等。

A group is an algebraic structure capturing the abstract properties of the collection of symmetries of a geometric object or of a physical or mechanical system. The project will concentrate on the study of free-by-cyclic groups. These groups are natural counterparts of the fundamental groups of fibered 3-manifolds, which are important mathematical objects playing key roles in Topology, Differential Geometry, Ergodic Theory, Number Theory, Computational Complexity, and other branches of Mathematics. Compared with the 3-manifold groups, the free-by-cyclic groups are much less well understood, even though they exhibit a considerably wider variety of interesting features and behavior. The project aims to study the geometry and dynamics of free-by-cyclic groups, particularly through exploring new polynomial invariants of these groups. This research should help better understand both the differences and the similarities between the free-by-cyclic groups and their 3-manifold "cousins", and lead to new insights about the interactions of algebra and dynamics in group theory.The project will build on the new work of the proposer, joint with Leininger and Dowdall, studying the properties of free-by-cyclic groups by means of a "folded mapping torus", associated to such a group. The folded mapping torus comes equipped with a natural semi-flow of continuous transformations, and it encodes in a particularly useful and accessible form key algebraic, geometric and dynamical information about the group. Another key object, associated to the folded mapping torus, is the "McMullen polynomial", which is a free-by-cyclic analog of the Teichmuller polynomial for fibered 3-manifolds. The main overall research goal of the project is to understand algebraic, geometric and dynamical features related to the many possible ways in which a free-by-cyclic group G can split as a free-by-cyclic group and as an ascending HNN-extension of a free group, and to gain new knowledge about free group automorphisms. Specific problems to be studied include: understanding the relationship between the "cone of sections" of folded mapping torus with the BNS-invariant of the group, uniformizing the stable trees and the Cannon-Thurston maps associated with different splittings of G, studying irreducibility properties of monodromy automorphisms associated to different free-by-cyclic splittings of G, developing an analog of the homological zeta-function for G, and others.