Related papers: Uniformly exponential growth and mapping class gro…
The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable…
Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes. Associated with grid class $\mathrm{Grid}(M)$ is a graph, $G(M)$, known…
The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering space in the mapping class group of the total surface. We give finite presentations for the balanced…
We construct finite-dimensional projective representations of the mapping class groups of compact connected oriented surfaces having one boundary component using stated skein algebras.
Under the natural action of the pure mapping class group of a surface of genus at least three, we show that any global fixed point in the low-dimensional deformation space of the surface group corresponds to the trivial representation. A…
In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various…
We prove, for various important classes of Mealy automata, that almost all generated groups have an element of infinite order. In certain cases, it also implies other results such as exponential growth.
This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic…
We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type…
We prove that a finitely generated solvable group which is not virtually nilpotent has exponential conjugacy growth.
This paper introduces iterated monodromy groups for transcendental functions and discusses them in the simplest setting, for post-singularly finite exponential functions. These groups are self-similar groups in a natural way, based on an…
We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in…
We consider the growth of heights of the points of the orbits of (piecewise) affine maps of the plane, with rational parameters. We analyse the asymptotic growth rate of both global and local ($p$-adic) heights, for the primes $p$ that…
It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping…
We use actions on trees to determine uniform exponential growth for subgroups of $GL_2$.
In this mostly expository article, we provide a new account of our proof with Minsky and Sisto that mapping class groups and Teichm\"uller spaces admit bicombings. More generally, we explain how the hierarchical hull of a pair of points in…
In this paper, we consider mappings on uniform domains with exponentially integrable distortion whose Jacobian determinants are integrable. We show that such mappings can be extended to the boundary and moreover these extensions are…
A profinite group equipped with an expansive endomorphism is equivalent to a one-sided group shift. We show that these groups have a very restricted structure. More precisely, we show that any such group can be decomposed into a finite…
We prove that the extended mapping class group is generated by three orientation reversing involutions.
This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded by means of the topological monodromy representation. Building off of prior work of the authors, we show that the…