# McShane identities for Higher Teichm\"uller theory and the Goncharov-Shen potential

@article{Huang2019McShaneIF, title={McShane identities for Higher Teichm\"uller theory and the Goncharov-Shen potential}, author={Yi Huang and Zhe Sun}, journal={arXiv: Geometric Topology}, year={2019} }

In [GS15], Goncharov and Shen introduce a family of mapping class group invariant regular functions on their $\mathcal{A}$-moduli space to explicitly formulate a particular homological mirror symmetry conjecture. Using these regular functions, we obtain McShane identities general rank positive surface group representations with loxodromic boundary monodromy and (non-strict) McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are…

## 19 Citations

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## References

SHOWING 1-10 OF 96 REFERENCES

### Basmajian's identity in higher Teichmüller–Thurston theory

- Mathematics
- 2015

We prove an extension of Basmajian's identity to n ‐Hitchin representations of compact bordered surfaces. For n=3 , we show that this identity has a geometric interpretation for convex real…

### McShane’s identity for classical Schottky groups

- Mathematics
- 2004

In [15], Greg McShane demonstrated a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. This was generalized by the authors in [19] to hyperbolic…

### Moduli spaces of local systems and higher Teichmüller theory

- Mathematics
- 2003

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S…

### McShane-Type Identities for Quasifuchsian Representations of Nonorientable Surfaces

- Mathematics
- 2018

We adapt Bers' double uniformization for nonorientable surfaces and show that the space $\mathcal{QF}(N)$ of quasifuchsian representations for a nonorientable surface $N$ is the Teichm\"uller space…

### The decorated Teichmüller space of punctured surfaces

- Mathematics
- 1987

A principal ℝ+5-bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several…

### Markoff triples and quasifuchsian groups

- Mathematics
- 1998

We study the global behaviour of trees of Markoff triples over the complex numbers. We relate this to the space of type‐preserving representations of the punctured torus group into SL(2,C). In…

### Simple geodesics and Markoff quads

- Mathematics
- 2013

The action of the mapping class group of the thrice-punctured projective plane on its $$\mathop {\mathrm{GL}}\nolimits (2,{\mathbb {C}})$$GL(2,C) character variety produces an algorithm for…

### Group completions and limit sets of Kleinian groups

- Mathematics
- 1980

Finitely generated groups and actions of finitely generated groups often come up in studying topology and geometry. While the most important example may be as fundamental groups of compact manifolds,…

### RANK n SWAPPING ALGEBRA FOR PGLn FOCK–GONCHAROV X MODULI SPACE

- Mathematics
- 2019

The rank n swapping algebra is a Poisson algebra defined on the set of ordered pairs of points of the circle using linking numbers, whose geometric model is given by a certain subspace of (Kn ×…

### Parameterizing Hitchin components

- Mathematics
- 2014

We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and…