We prove that if $ϕ,ψ\in Out(F_N)$ are hyperbolic iwips (irreducible with irreducible powers) such that $<ϕ,ψ>\le Out(F_N)$ is not virtually cyclic then some high powers of $ϕ$ and $ψ$ generate a free subgroup of rank two, all of whose nontrivial elements are again hyperbolic iwips. Being a hyperbolic iwip element of $Out(F_N)$ is strongly analogous to being a pseudo-Anosov element of a mapping class group, so the above result provides analogs of "purely pseudo-Anosov" free subgroups of $Out(F_N)$.
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.
We adapt the Ping-Pong Lemma, which historically was used to study free products of groups, to the setting of the homeomorphism group of the unit interval. As a consequence, we isolate a large class of generating sets for subgroups of $\mathrm{Homeo}_+(I)$ for which certain finite dynamical data can be used to determine the marked isomorphism type of the groups which they generate. As a corollary, we will obtain a criteria for embedding subgroups of $\mathrm{Homeo}_+(I)$ into Richard Thompson's group $F$. In particular, every member of our class of generating sets generates a group which embed…
We discuss actions of free groups on the circle with "ping-pong" dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group $F_n$ admits an isolated circular order if and only if n is even, in stark contrast with the case for linear orders. This answers a question from (Mann, Rivas, 2016). Inspired by work of Alvarez, Barrientos, Filimonov, Kleptsyn, Malicet, Menino and Triestino, we also exhibit examples of "exotic" isolated points in the space of all circular orders on $F_2$. An…
Let $G$ be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex $X$ without fixed points at infinity. We show that for any finite collection of simultaneously inessential subgroups $\{H_1, \ldots, H_k\}$ in $G$, there exists an element $g$ of infinite order such that $\forall i$, $\langle H_i, g\rangle \cong H_i * \langle g\rangle$. We apply this to show that any group, acting faithfully and geometrically on a non-Euclidean possibly reducible CAT(0) cube complex, has property $P_{naive}$ i.e. given any finite list $\{g_1, \ldots, g_k\}…
Ping-pong in the projective plane over a nonarchimedean field
Mathematisches Institut der Universität Bonn（波恩大学数学研究所）；Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎-萨克雷大学，法国国家科学研究中心，奥赛数学实验室）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数学研究所）
We show that for any $n\geq 2$, two elements selected uniformly at random from a \emph{symmetrized} Euclidean ball of radius $X$ in $\textrm{SL}_n(\mathbb Z)$ will generate a thin free group with probability tending to $1$ as $X\rightarrow \infty.$ This is done by showing that the two elements will form a ping-pong pair, when acting on a suitable space, with probability tending to $1$. On the other hand, we give an upper bound less than $1$ for the probability that two such elements will form a ping-pong pair in the usual Euclidean ball model in the case where $n>2$.
We exhibit an infinite family of discrete subgroups of ${Sp}_4(\mathbb R)$ which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters $\left(\tfrac{N-3}{2N},\tfrac{N-1}{2N}, \tfrac{N+1}{2N}, \tfrac{N+3}{2N}\right)$ at infinity and maximal unipotent monodromy at zero, for any integer $N\geq 4$. Additionally, we relate the cones used for ping-pong in $\mathbb R^4$ with crooked surfaces, which we then use to exhibit do…
Using a theorem of J. Groves we give a ping-pong proof of Osin's uniform exponential growth for solvable groups. We discuss slow exponential growth and show that this phenomenon disappears as one passes to a finite index subgroup.
We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, \dots, h_k$, there exists another element $γ\neq 1$ such that for all $i$, $\langle h_i, γ\rangle = \langle h_i \rangle* \langle γ\rangle$. We also obtain that if a collection of subgroups $H_1, \dots, H_k$ is a hyperbolically embedded collection, then there is $γ\neq 1$ such that for all $i$, $\langle H_i, γ\rangle = H_i * \langle γ\rangle$.
We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be either the whole mapping class group, a direct product of a free group of rank two with the cyclic group of order two, or a free group of rank at most three. We characterize subgroups generated by a collection of (respectively squares of) Dehn twists, if there is a pair among so that …
This is the fourth and last in a series of four papers (with research announcement posted on this arXiv) that develop a decomposition theory for subgroups of $\text{Out}(F_n)$. In this paper we develop general ping-pong techniques for the action of $\text{Out}(F_n)$ on the space of lines of $F_n$. Using these techniques we prove the main results stated in the research announcement, Theorem C and its special case Theorem I, the latter of which says that for any finitely generated subgroup $\mathcal H$ of $\text{Out}(F_n)$ that acts trivially on homology with $\mathbb{Z}/3$ coefficients, and for…
We present a practical algorithm which, given a non-archimedean local field $K$ and any two elements $A,B\in {\rm SL_2}(K)$, determines after finitely many steps whether or not the subgroup $\langle A, B \rangle\le {\rm SL_2}(K)$ is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the action of ${\rm SL_2}(K)$ by isometries on its Bruhat-Tits tree. The algorithm itself can also be used for two-generated subgroups of the isometry group of any locally finite simplicial tree, and has applications to the constructive membership problem. In an appendix joint with Fréd…
We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be rel…
