We exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was
initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that $BS(1,n)$ has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].
For an element in $BS(1,n) = langle t,a | tat^{-1} = a^n rangle$ written in the normal form $t^{-u}a^vt^w$ with $u,w geq 0$ and $v in mathbb{Z}$, we exhibit a geodesic word representing the element and give a formula for its word length with respect
to the generating set ${t,a}$. Using this word length formula, we prove that there are sets of elements of positive density of positive, negative and zero conjugation curvature, as defined by Bar Natan, Duchin and Kropholler.
Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of th
e paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.
ReLU neural networks define piecewise linear functions of their inputs. However, initializing and training a neural network is very different from fitting a linear spline. In this paper, we expand empirically upon previous theoretical work to demonst
rate features of trained neural networks. Standard network initialization and training produce networks vastly simpler than a naive parameter count would suggest and can impart odd features to the trained network. However, we also show the forced simplicity is beneficial and, indeed, critical for the wide success of these networks.
The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor
set. As a consequence, we prove that the Gromov boundary of the ray graph is homeomorphic to a quotient of a subset of the circle. This version contains some updates and corrections.
In 1985, Barnsley and Harrington defined a ``Mandelbrot Set $mathcal{M}$ for pairs of similarities --- this is the set of complex numbers $z$ with $0<|z|<1$ for which the limit set of the semigroup generated by the similarities $x mapsto zx$ and $x m
apsto z(x-1)+1$ is connected. Equivalently, $mathcal{M}$ is the closure of the set of roots of polynomials with coefficients in $lbrace -1,0,1 rbrace$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ``holes in $mathcal{M}$, and conjectured that these holes were genuine. These holes are very interesting, since they are ``exotic components of the space of (2 generator) Schottky semigroups. The existence of at least one hole was rigorously confirmed by Bandt in 2002, and he conjectured that the interior points are dense away from the real axis. We introduce the technique of traps to construct and certify interior points of $mathcal{M}$, and use them to prove Bandts Conjecture. Furthermore, our techniques let us certify the existence of infinitely many holes in $mathcal{M}$.
We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surfa
ce. In the case that the orbifold is a disk, there are some conditions. Our results generalize work of Calegari-Louwsma and resolve a conjecture of Calegari.
We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.
