The Hurwitz graph has as its vertices the maximal chains in the lattice of non-crossing partitions, with an edge between two chains which differ at exactly one place. Adin and Roichman introduced an order on this graph by choosing a base element and showed that the maximal elements are counted by the Catalan numbers. We show that this set of maximal elements is in natural bijection with non-crossing partitions so that the graph structure corresponds to that of the permutahedron, restricted to non-crossing partitions. We use this to compute the diameter of the graph of maximal elements. We also briefly consider the graph obtained by identifying maximal chains with minimal factorizations of a full cycle into transpositions in the symmetric groups and quotienting by commutation relations.
We establish two rigidity theorems for non-left-orderable group actions on dendrites. Explicitly, letting X be a dendrite without infinite order points, we show that if Gamma is a non-virtually left-orderable small group, then no Gamma-action on X is almost free, and if Gamma is a higher-rank lattice, then every Gamma-action on X is a highly proximal extension of an almost finite action.
In 2021, Belk, Bleak and Matucci introduced a symbolic coding of the horofunction boundary of a hyperbolic group as a clopen subset of a shift of finite type. This coding was first used by them that same year to show that hyperbolic groups embed into rational groups and then by the same authors together with Zaremsky (2023) to embed them into finitely presented simple groups (solving the Boone-Higman conjecture in the hyperbolic case). In this paper, we provide an explicit description of the Gromov boundary as a quotient of horofunction boundary based on this symbolic coding and deduce several geometric consequences. In particular, we construct a synchronous automaton which determines whether two horofunctions map to the same point in the Gromov boundary.
We show that the Cohen–Lyndon property holds for all non-metric small cancellation quotients. This generalises the analogous result from the metric small cancellation setting and answers a question asked by Lyndon in 1966 and by Wall in his 1979 problem list.
In this paper, we study the mapping class group action on the relative \operatorname{PSL}(2,\mathbb{R}) -character varieties of punctured surfaces. It is well known that Minsky’s primitive-stable representations form a domain of discontinuity for the \operatorname{Out}(F_{n}) -action on the \operatorname{PSL}(2,\mathbb{C}) -character variety. We define simple stability of representations of fundamental group of a surface into \operatorname{PSL}(2,\mathbb{R}) which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the \operatorname{MCG} -action. Our first main result shows that holonomies of hyperbolic cone surfaces are simple-stable. We also prove that holonomies of hyperbolic cone surfaces with exactly one cone-point of cone-angle less than \pi are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations.
