Related papers: Combinatorial Growth in the Modular Group

The m-thick part of the modular surface X is the smallest compact subsurface of X with horocycle boundary containing all the closed geodesics which wind around the cusp at most m times. The m-thick parts form a compact exhaustion of X. We…

We prove exponential growth rate of contractible closed geodesics for an arbitrary bumpy metric on manifolds of the form X#Y, where the fundamental group of X has a subgroup of finite index at least 3 and Y is simply connected and not a…

We prove the existence of infinitely many low-lying and fundamental closed geodesics on the modular surface which are reciprocal, that is, invariant under time reversal. The method combines ideas from Parts I and II of this series, namely…

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing…

We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in PSL(2,R), generalizing earlier work of Sarnak and Bourgain-Kontorovich on the growth of the number of reciprocal geodesics on the modular…

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper…

We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group,…

This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely…

Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric…

We consider the functions that bound the dimensions of finite-dimensional associative or Lie algebras in terms of the dimensions of their commutative subalgebras. It is proved that these functions have quadratic growth. As a result, we also…

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as…

Let G denote a closed, connected, self adjoint, noncompact subgroup of GL(n,R), and let d_{R} denote the canonical right invariant Riemannian metric on G. For v in R^{n} let G_{v} = {g in G : g(v) = v}. We obtain algebraically defined upper…

Molino's description of Riemannian foliations on compact manifolds is generalized to the setting of compact equicontinuous foliated spaces, in the case where the leaves are dense. In particular, a structural local group is associated to…

Let G be a finitely generated group, let $\pi \colon G \to {\rm GL}(E)$ be a uniformly bounded $c_0$-representation on a superreflexive Banach space $E$, and let $b \colon G \to E$ be a $1$-cocycle for $\pi$. Then $b$ has sublinear growth…

We investigate the growth of Fourier coefficients of Siegel paramodular forms built by exponentially lifting weak Jacobi forms, focusing on terms with large negative discriminant. To this end we implement a method based on deforming…

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For…

Every finite collection of oriented closed geodesics in the modular surface has a canonically associated link in its unit tangent bundle coming from the periodic orbits of the geodesic flow. We study the volume of the associated link…

Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus g endowed with the Weil-Petersson metric. In this paper, we introduce a function $L(g)$ of genus $g$ and call the geodesics whose length less than $L(g)$ short…

In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the…

It is a conjecture of Colin and Honda that the number of Reeb periodic orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact…