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We derive generalizations of McShane’s identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen, which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman–Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.

Let $n\\geq 1$, and let $\\iota_{n}\\colon\\thinspace F_{n}(M) \\longrightarrow \\prod_{1}^{n}\\, M$ be the natural inclusion of the $n$th configuration space of $M$ in the $n$-fold Cartesian product of $M$ with itself. In this paper, we study the map $\\iota_{n}$, the homotopy fibre $I_{n}$ of $\\iota_{n}$ and its homotopy groups, and the induced homomorphisms $(\\iota_{n})_{\\#k}$ on the $k$th homotopy groups of $F_{n}(M)$ and $\\prod_{1}^{n}\\, M$ for all $k\\geq 1$, where $M$ is the $2$-sphere $\\mathbb{S}^{2}$ or the real projective plane $\\mathbb{R}P^{2}$. It is well known that the group $\\pi_{k}(I_{n})$ is the homotopy group $\\pi_{k+1}(\\prod_{1}^{n}\\, M, F_n(M))$ for all $k\\geq 0$. If $k\\geq 2$, we show that the homomorphism $(\\iota_{n})_{\\#k}$ is injective and diagonal, with the exception of the case $n=k=2$ and $M=\\mathbb{S}^{2}$, where it is anti-diagonal. We then show that $I_{n}$ has the homotopy type of $K(R_{n-1},1) \\times \\Omega(\\prod_{1}^{n-1} \\mathbb{S}^{2})$, where $R_{n-1}$ is the $(n-1)$th Artin pure braid group if $M=\\mathbb{S}^{2}$, and is the fundamental group $G_{n-1}$ of the $(n-1)$th orbit configuration space of the open cylinder $\\mathbb{S}^{2} \\setminus \\{\\widetilde{z}_{0}, -\\widetilde{z}_{0}\\}$ with respect to the action of the antipodal map of $\\mathbb{S}^{2}$ if $M=\\mathbb{R}P^{2}$, where $\\widetilde{z}_{0}\\in \\mathbb{S}^{2}$. This enables us to describe the long exact sequence in homotopy of the homotopy fibration $I_{n} \\longrightarrow F_n(M)\\stackrel{\\iota_{n}}{\\longrightarrow} \\prod_{1}^{n}\\, M$ in geometric terms, and notably the image of the boundary homomorphism $\\pi_{k+1}(\\prod_{1}^{n}\\, M)\\longrightarrow \\pi_{k}(I_{n})$. From this, if $M=\\mathbb{S}^{2}$ and $n\\geq 3$ (resp. $M=\\mathbb{R}P^{2}$ and $n\\geq 2$), we show that $\\ker{(\\iota_{n})_{\\#1}}$ is isomorphic to the quotient of $R_{n-1}$ by the square of its centre, as well as to an iterated semi-direct product of free groups with the subgroup of order $2$ generated by the centre of $P_{n}(M)$ that is reminiscent of the combing operation for the Artin pure braid groups, as well as decompositions obtained in a previous paper.This paper is a shortened version of \"The homotopy fibre of the inclusion $F_{n}(M) \\longrightarrow \\prod_{1}^{n}\\, M$ for $M$ either $\\mathbb{S}^{2}$ or the real projective plane $\\mathbb{R}P^{2}$ and orbit configuration spaces\", ⟨hal-01627001⟩. The main difference between the two is the statement and proof of Theorem 3.

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the latter one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups,

In this survey we explore relationships between several different representations of braid groups as automorphisms of free groups as well as induced linear representations.

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.

The interplay between topological hyperconvex spaces and sigma-finite measures in such spaces gives rise to a set of analytical observations. This paper introduces the Noetherian class of k-finite k-hyperconvex topological subspaces (NHCs) admitting countable finite covers. A sigma-finite measure is constructed in a sigma-semiring in a NHC under a topological ordering of NHCs. The topological ordering relation maintains the irreflexive and anti-symmetric algebraic properties while retaining the homeomorphism of NHCs. The monotonic measure sequence in a NHC determines the convexity and compactness of topological subspaces. Interestingly, the topological ordering in NHCs in two isomorphic topological spaces induces the corresponding ordering of measures in sigma-semirings. Moreover, the uniform topological measure spaces of NHCs need not always preserve the pushforward measures, and a NHC semiring is functionally separable by a set of inner-measurable functions