Related papers: A remark on the word length in surface groups

In this note, we connect the $n$-torsions of the Picard group of an elliptic surface to the $n$-divisibility of the class group of torsion fields for a given integer $n>1$. We also connect the $n$-divisibility of the Selmer group to that of…

A word in a free group is called ``potentially positive'' if it is automorphic to an element which is written with only positive exponents. We will develop automata to analyze properties of potentially positive words. We will use these to…

We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded…

Given a finitely generated group with generating set $S$, we study the \emph{cogrowth} sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks…

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing…

This article presents a combinatorial result on indexed languages which was inspired by an attempt to understand the structure of groups with indexed language word problem. We show that a sufficiently long word in an indexed language can be…

For an orientable surface $S$ of finite topological type with genus $g \geq 3$, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of $S$. The set constructed, and the method of rigid expansion,…

Let $\phi \in {\rm Mod}(\Sigma)$ be an arbitrary element of the mapping class group of a closed orientable surface $\Sigma$ of genus at least $2$. For any characteristic cover $\widetilde{\Sigma} \to \Sigma$ one can consider the linear…

We introduce a numerical invariant $\zeta(\Sigma)$ measuring the end-complexity of $\Sigma$ and use it to organize coarse-geometric features of Map($\Sigma$). Our main tool is the \emph{non-peripheral curve graph} $C_{\rm np}(\Sigma)$,…

We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also…

For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of its integer points. We give an interpretation of these…

We present a new short proof of the explicit formula for the group of links (and also link maps) in the 'quadruple point free' dimension. Denote by $L^m_{p,q}$ (respectively, $C^{m-p}_p$) the group of smooth embeddings $S^p\sqcup S^q\to…

The filling length of an edge-circuit \eta in the Cayley 2-complex of a finite presentation of a group is the least integer L such that there is a combinatorial null-homotopy of \eta down to a base point through loops of length at most L.…

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed…

We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to…

We show that for any finite-rank free group $\Gamma$, any word-equation in one variable of length $n$ with constants in $\Gamma$ fails to be satisfied by some element of $\Gamma$ of word-length $O(\log (n))$. By a result of the first…

The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric.…

Let $\Sigma_{g,p}$ be a oriented connected surface of genus $g$ with $p$ punctures. We denote by $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ the mapping class group and the extended mapping class group of $\Sigma_{g,p}$, respectively.…

A perforated surface is the complement $\mathring\Sigma:=\Sigma\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\Sigma$. It is known that the topological type of $\Sigma\setminus A$ is independent of the…

Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive…