of the standard form

$$ \langle a_1, b_1, \dots, a_g, b_g, x_1, \dots, x_k \mid [a_1, b_1] \cdots [a_g, b_g] x_1 \cdots x_k = 1 = x_1^{m_1} = \cdots = x_k^{m_k} \rangle. \tag{A.1} $$

We can define the *expected dimension* $\dim_e \mathrm{Hit}(\pi_1 Y, \mathbf{PGL}(n, \mathbb{R}))$ of the Hitchin component considering that for each $i = 1, \dots, g$, the generators $a_i$ and $b_i$ can be mapped to elements of $\mathbf{PGL}(n, \mathbb{R})$ that form an open subset (see Proposition 2.20), so we add the dimension $\dim \mathbf{PGL}(n, \mathbb{R})$ for each one of them. For each $j = 1, \dots, k$, instead, we have seen in the proof of Proposition 2.18 that the generator $x_j$ can be mapped to an element of $\mathbf{PGL}(n, \mathbb{R})$ which is conjugate to $\kappa(\tau_m)$, where $\kappa$ is the principal representation, as in (2.3), and $\tau_m$ is the matrix

$$ \begin{bmatrix} \cos(\pi/m) & -\sin(\pi/m) \\ \sin(\pi/m) & \cos(\pi/m) \end{bmatrix} \in \mathbf{PGL}(2, \mathbb{R}). $$

Let $\mathcal{D}_n(\mathbb{Z}/m\mathbb{Z})$ be the component of the representation variety $\mathrm{Hom}(\mathbb{Z}/m\mathbb{Z}, \mathbf{PGL}(n, \mathbb{R}))$ containing the representation $\rho$ given by $\rho(1) = \kappa(\tau_m)$. For every $x_i$, we add the dimension $\dim \mathcal{D}_n(\mathbb{Z}/m_j\mathbb{Z})$. We also need to consider the relation $[a_1, b_1] \cdots [a_g, b_g] x_1 \cdots x_k = 1$, so we subtract a term equal to $\dim \mathbf{PGL}(n, \mathbb{R})$. Finally, since the Hitchin component is the space of conjugacy classes of Hitchin representations, we again subtract $\dim \mathbf{PGL}(n, \mathbb{R})$. In this way we obtain the *expected dimension* of the Hitchin components:

$$ \dim_e \mathrm{Hit}(\pi_1 Y, \mathbf{PGL}(n, \mathbb{R})) := (2g - 2) \dim \mathbf{PGL}(n, \mathbb{R}) + \sum_{i=1}^{k} \dim \mathcal{D}_n(\mathbb{Z}/m_i\mathbb{Z}). $$

The arguments used to define the expected dimension are just a way to guess the dimension, but they do not constitute a proof that the actual dimension is the same. We will now prove that this expected dimension agrees with the actual dimension. We note that Theorem 5.8 determines not only the dimension, but also the topology of the Hitchin component.

**Remark A.1.** We can similarly define the expected dimension of other components of the representation space $\mathrm{Hom}(\pi_1 Y, \mathbf{PGL}(n, \mathbb{R}))/\mathbf{PGL}(n, \mathbb{R})$. However, it may not be true that the expected dimension is in fact equal to the actual dimension in those cases. In fact, the expected dimension can sometimes be negative [6]. That is why, in this paper, we do not mention the expected dimensions of components of $\mathrm{Rep}(\pi_1 Y, \mathbf{PGL}(n, \mathbb{R}))$ other than the Hitchin component.

**Proposition A.2.** One has $\dim_e \mathrm{Hit}(\pi_1 Y, \mathbf{PGL}(n, \mathbb{R})) = \dim_e \mathrm{Hit}(\pi_1 Y, \mathbf{PGL}(n, \mathbb{R}))$.

*Proof.* Since we know from Theorem 5.8 that the Hitchin component is of dimension $(2g - 2) \dim \mathbf{PGL}(n, \mathbb{R}) + 2 \sum_{d=2}^n \sum_{i=1}^k O(d, m_i)$, the only thing we need to show is that $\dim \mathcal{D}_n(\mathbb{Z}/m\mathbb{Z}) = 2 \sum_{d=2}^n O(d, m)$. In [51], Long and Thistlethwaite introduced an arithmetic function of two variables $\sigma(n, m)$ for $n, m \ge 2$, and showed that $\mathcal{D}_n(\mathbb{Z}/m\mathbb{Z})$
is of dimension $n^2 - \sigma(n, m)$. Here, if $q$ and $r$ are the quotient and remainder of dividing $n$ by $m$, respectively, i.e. $n = mq + r$ with $0 \le r < m$, then $\sigma(n, m) = (n + r)q + r$. Then the lemma is a consequence of the following simple computation:

$$ 
\begin{aligned}
2 \sum_{d=2}^{n} O(d, m) &= 2 \sum_{d=1}^{n} \left( d + \lfloor -\frac{d}{m} \rfloor \right) = 2 \sum_{d=1}^{n} d - 2 \sum_{d=1}^{mq+r} \lfloor \frac{d}{m} \rfloor \\
&= 2 \cdot \frac{n(n+1)}{2} - 2 \left( m \cdot \frac{q(q+1)}{2} + r(q+1) \right) = n^2 - \sigma(n, m). \quad \blacksquare
\end{aligned}
$$

## Appendix B. Tables

<table><caption>**Tab. B.1.** The dimension and exponents of simple complex Lie algebras.</caption><thead><tr><th>Int(g<sub>C</sub>)<sup>τ</sup></th><th>Dimension</th><th>Rank</th><th>Exponents</th></tr></thead><tbody><tr><td>PGL(n, R) (n ≥ 2)</td><td>n<sup>2</sup> - 1</td><td>n - 1</td><td>1, 2, ..., n - 1</td></tr><tr><td>PSp<sup>±</sup>(2m, R) (m ≥ 1)</td><td>m(2m + 1)</td><td>m</td><td>1, 3, ..., 2m - 1</td></tr><tr><td>PO(m, m + 1) (m ≥ 1)</td><td>m(2m + 1)</td><td>m</td><td>1, 3, ..., 2m - 1</td></tr><tr><td>PO<sup>±</sup>(m, m) (m ≥ 3)</td><td>m(2m - 1)</td><td>m</td><td>1, 3, ..., 2m - 3; m - 1</td></tr><tr><td>G<sub>2</sub></td><td>14</td><td>2</td><td>1, 5</td></tr><tr><td>F<sub>4</sub></td><td>52</td><td>4</td><td>1, 5, 7, 11</td></tr><tr><td>E<sub>6</sub></td><td>78</td><td>6</td><td>1, 4, 5, 7, 8, 11</td></tr><tr><td>E<sub>7</sub></td><td>133</td><td>7</td><td>1, 5, 7, 9, 11, 13, 17</td></tr><tr><td>E<sub>8</sub></td><td>248</td><td>8</td><td>1, 7, 11, 13, 17, 19, 23, 29</td></tr></tbody></table>

<table><caption>**Tab. B.2.** Dimension of Hitchin components for groups of rank ≤ 6.</caption><thead><tr><th>Group</th><th>Dimension of Hit(π<sub>1</sub> Y, G) for Y closed</th></tr></thead><tbody><tr><td>PGL(2, R)</td><td>-3χ(|Y|) + 2k + l [62]</td></tr><tr><td>PGL(3, R)</td><td>-8χ(|Y|) + (6k - 2k<sub>2</sub>) + (3l - l<sub>2</sub>) [9]</td></tr><tr><td>PGL(4, R)</td><td>-15χ(|Y|) + (12k - 4k<sub>2</sub> - 2k<sub>3</sub>) + (6l - 2l<sub>2</sub> - l<sub>3</sub>)</td></tr><tr><td>PGL(5, R)</td><td>-24χ(|Y|) + (20k - 8k<sub>2</sub> - 4k<sub>3</sub> - 2k<sub>4</sub>) + (10l - 4l<sub>2</sub> - 2l<sub>3</sub> - l<sub>4</sub>)</td></tr><tr><td>PGL(6, R)</td><td>-35χ(|Y|) + (30k - 12k<sub>2</sub> - 6k<sub>3</sub> - 4k<sub>4</sub> - 2k<sub>5</sub>) + (15l - 6l<sub>2</sub> - 3l<sub>3</sub> - 2l<sub>4</sub> - l<sub>5</sub>)</td></tr><tr><td>PGL(7, R)</td><td>-48χ(|Y|) + (42k - 18k<sub>2</sub> - 10k<sub>3</sub> - 6k<sub>4</sub> - 4k<sub>5</sub> - 2k<sub>6</sub>) + (21l - 9l<sub>2</sub> - 5l<sub>3</sub> - 3l<sub>4</sub> - 2l<sub>5</sub> - l<sub>6</sub>)</td></tr><tr><td>PSp<sup>±</sup>(4, R)</td><td>-10χ(|Y|) + (8k - 2k<sub>2</sub> - 2k<sub>3</sub>) + (4l - l<sub>2</sub> - l<sub>3</sub>)</td></tr><tr><td>PSp<sup>±</sup>(6, R)</td><td>-21χ(|Y|) + (18k - 6k<sub>2</sub> - 4k<sub>3</sub> - 2k<sub>4</sub> - 2k<sub>5</sub>) + (9l - 3l<sub>2</sub> - 2l<sub>3</sub> - l<sub>4</sub> - l<sub>5</sub>)</td></tr><tr><td>PSp<sup>±</sup>(8, R)</td><td>-36χ(|Y|) + (32k - 12k<sub>2</sub> - 8k<sub>3</sub> - 4k<sub>4</sub> - 4k<sub>5</sub> - 2k<sub>6</sub> - 2k<sub>7</sub>) + (16l - 6l<sub>2</sub> - 4l<sub>3</sub> - 2l<sub>4</sub> - 2l<sub>5</sub> - l<sub>6</sub> - l<sub>7</sub>)</td></tr><tr><td>PSp<sup>±</sup>(10, R)</td><td>-55χ(|Y|) + (50k - 20k<sub>2</sub> - 14k<sub>3</sub> - 8k<sub>4</sub> - 6k<sub>5</sub> - 4k<sub>6</sub> - 4k<sub>7</sub> - 2k<sub>8</sub> - 2k<sub>9</sub>) + (25l - 10l<sub>2</sub> - 7l<sub>3</sub> - 4l<sub>4</sub> - 3l<sub>5</sub> - 2l<sub>6</sub> - 2l<sub>7</sub> - l<sub>8</sub> - l<sub>9</sub>)</td></tr><tr><td>PSp<sup>±</sup>(12, R)</td><td>-78χ(|Y|) + (72k - 30k<sub>2</sub> - 20k<sub>3</sub> - 12k<sub>4</sub> - 10k<sub>5</sub> - 6k<sub>6</sub> - 6k<sub>7</sub> - 4k<sub>8</sub> - 4k<sub>9</sub> - 2k<sub>10</sub> - 2k<sub>11</sub>) + (36l - 15l<sub>2</sub> - 10l<sub>3</sub> - 6l<sub>4</sub> - 5l<sub>5</sub> - 3l<sub>6</sub> - 3l<sub>7</sub> - 2l<sub>8</sub> - 2l<sub>9</sub> - l<sub>10</sub> - l<sub>11</sub>)</td></tr><tr><td>PO<sup>±</sup>(4, 4)</td><td>-28χ(|Y|) + (24k - 8k<sub>2</sub> - 6k<sub>3</sub> - 2k<sub>4</sub> - 2k<sub>5</sub>) + (12l - 4l<sub>2</sub> - 3l<sub>3</sub> - l<sub>4</sub> - l<sub>5</sub>)</td></tr><tr><td>PO<sup>±</sup>(5, 5)</td><td>-45χ(|Y|) + (40k - 16k<sub>2</sub> - 10k<sub>3</sub> - 6k<sub>4</sub> - 4k<sub>5</sub> - 2k<sub>6</sub> - 2k<sub>7</sub>) + (20l - 8l<sub>2</sub> - 5l<sub>3</sub> - 3l<sub>4</sub> - 2l<sub>5</sub> - l<sub>6</sub> - l<sub>7</sub>)</td></tr><tr><td>PO<sup>±</sup>(6, 6)</td><td>-66χ(|Y|) + (60k - 24k<sub>2</sub> - 16k<sub>3</sub> - 10k<sub>4</sub> - 8k<sub>5</sub> - 4k<sub>6</sub> - 4k<sub>7</sub> - 2k<sub>8</sub> - 2k<sub>9</sub>) + (30l - 12l<sub>2</sub> - 8l<sub>3</sub> - 5l<sub>4</sub> - 4l<sub>5</sub> - 2l<sub>6</sub> - 2l<sub>7</sub> - l<sub>8</sub> - l<sub>9</sub>)</td></tr></tbody></table>
<table><caption>Tab. B.2 (cont.)</caption><thead><tr><th>Group</th><th>Dimension of Hit(π<sub>1</sub>Y, G) for Y closed</th></tr></thead><tbody><tr><td>G<sub>2</sub></td><td>-14χ(|Y|)+(12k-4k<sub>2</sub>-2k<sub>3</sub>-2k<sub>4</sub>-2k<sub>5</sub>)+(6ℓ-2ℓ<sub>2</sub>-ℓ<sub>3</sub>-ℓ<sub>4</sub>-ℓ<sub>5</sub>)</td></tr><tr><td>F<sub>4</sub></td><td>-52χ(|Y|)+(48k-20k<sub>2</sub>-12k<sub>3</sub>-8k<sub>4</sub>-8k<sub>5</sub>-4k<sub>6</sub>-4k<sub>7</sub>-2k<sub>8</sub>-2k<sub>9</sub>-2k<sub>10</sub>-2k<sub>11</sub>)<br/>+(24ℓ-10ℓ<sub>2</sub>-6ℓ<sub>3</sub>-4ℓ<sub>4</sub>-4ℓ<sub>5</sub>-2ℓ<sub>6</sub>-2ℓ<sub>7</sub>-ℓ<sub>8</sub>-ℓ<sub>9</sub>-ℓ<sub>10</sub>-ℓ<sub>11</sub>)</td></tr><tr><td>E<sub>6</sub></td><td>-78χ(|Y|)+(72k-32k<sub>2</sub>-18k<sub>3</sub>-14k<sub>4</sub>-10k<sub>5</sub>-6k<sub>6</sub>-6k<sub>7</sub>-4k<sub>8</sub>-2k<sub>9</sub>-2k<sub>10</sub>-2k<sub>11</sub>)<br/>+(36ℓ-16ℓ<sub>2</sub>-9ℓ<sub>3</sub>-7ℓ<sub>4</sub>-5ℓ<sub>5</sub>-3ℓ<sub>6</sub>-3ℓ<sub>7</sub>-2ℓ<sub>8</sub>-ℓ<sub>9</sub>-ℓ<sub>10</sub>-ℓ<sub>11</sub>)</td></tr></tbody></table>

<table><caption>Tab. B.3. Hitchin components containing only cyclic or (n-1)-cyclic Higgs bundles.</caption><thead><tr><th>Group</th><th>Degree</th><th>(m<sub>1</sub>,...,m<sub>k</sub>) with m<sub>i</sub> ≤ m<sub>i+1</sub>, ∑<sup>1</sup><sub>m<sub>i</sub></sub> < k-2</th></tr></thead><tbody><tr><td rowspan="2">PGL(3, R)</td><td>2</td><td>(2, 2, 2, 2, 2) or (2, 2, 2, m<sub>4</sub>)</td></tr><tr><td>3</td><td>(m<sub>1</sub>, m<sub>2</sub>, m<sub>3</sub>) with m<sub>1</sub> ≥ 3</td></tr><tr><td rowspan="2">PGL(4, R)</td><td>3</td><td>(3, 3, m<sub>3</sub>)</td></tr><tr><td>4</td><td>(2, m<sub>2</sub>, m<sub>3</sub>) with m<sub>2</sub> ≥ 4</td></tr><tr><td>PGL(5, R)</td><td>4</td><td>(2, 4, m<sub>3</sub>)</td></tr><tr><td>PGL(n, R) with n = 6, 7</td><td>6</td><td>(2, 3, m<sub>3</sub>)</td></tr><tr><td>PSp<sup>±</sup>(4, R) or PO(2, 3)</td><td>4</td><td>(m<sub>1</sub>, m<sub>2</sub>, m<sub>3</sub>) with m<sub>2</sub> ≠ 3</td></tr><tr><td>PSp<sup>±</sup>(6, R) or PO(3, 4)</td><td>6</td><td>(2, 3, m<sub>3</sub>) or (3, 3, m<sub>3</sub>)</td></tr><tr><td>G<sub>2</sub></td><td>6</td><td>(m<sub>1</sub>, m<sub>2</sub>, m<sub>3</sub>) with (m<sub>1</sub>, m<sub>3</sub>) ≠ (2, 5)</td></tr></tbody></table>

<table><caption>Tab. B.4. Spheres with k cone points satisfying H<sup>0</sup>(Y, K(Y, d)) = 0.</caption><thead><tr><th>k</th><th>d</th><th>(m<sub>1</sub>,...,m<sub>k</sub>) with m<sub>i</sub> ≤ m<sub>i+1</sub>, ∑<sup>1</sup><sub>m<sub>i</sub></sub> < k - 2</th></tr></thead><tbody><tr><td>2</td><td>(m<sub>1</sub>, m<sub>2</sub>, m<sub>3</sub>)</td><td></td></tr><tr><td>3</td><td>(2, m<sub>2</sub>, m<sub>3</sub>)</td><td></td></tr><tr><td>4</td><td>(2, 3, m<sub>3</sub>), (3, 3, m<sub>3</sub>)</td><td></td></tr><tr><td>5</td><td>(2, 3, m<sub>3</sub>), (2, 4, m<sub>3</sub>), (3, 3, 4), (3, 4, 4), (4, 4, 4)</td><td></td></tr><tr><td>6</td><td>(2, 4, 5), (2, 5, 5)</td><td></td></tr><tr><td>7</td><td>(2, 3, m<sub>3</sub>), (2, m<sub>2</sub>, m<sub>3</sub>) with 4 ≤ m<sub>2</sub> ≤ m<sub>3</sub> ≤ 6, (3, 3, m<sub>3</sub>) with 4 ≤ m<sub>3</sub> ≤ 6</td><td></td></tr><tr><td>8</td><td>(2, 3, 7)</td><td></td></tr><tr><td>9</td><td>(2, 3, 7), (2, 3, 8), (2, 4, m<sub>3</sub>) with 5 ≤ m<sub>3</sub> ≤ 8</td><td></td></tr><tr><td rowspan="5">3</td><td>10</td><td>(2, 3, m<sub>3</sub>) with 7 ≤ m<sub>3</sub> ≤ 9, (3, 3, 4)</td><td></td></tr><tr><td>11</td><td>(2, 3, m<sub>3</sub>) with 7 ≤ m<sub>3</sub> ≤ 10, (2, 4, 5), (2, 5, 5)</td><td></td></tr><tr><td>13</td><td>(2, 3, m<sub>3</sub>) with 7 ≤ m<sub>3</sub> ≤ 12, (2, 4, 5), (2, 4, 6), (3, 3, 4)</td><td></td></tr><tr><td>15, 16</td><td>(2, 3, 7)</td><td></td></tr><tr><td>17</td><td>(2, 3, 7), (2, 3, 8), (2, 4, 5)</td><td></td></tr><tr><td rowspan="5">4</td><td>19</td><td>(2, 3, m<sub>3</sub>) with 7 ≤ m<sub>3</sub> ≤ 9</td><td></td></tr><tr><td>21</td><td>(2, 4, 5)</td><td></td></tr><tr><td>22, 23</td><td>(2, 3, 7)</td><td></td></tr><tr><td>25</td><td>(2, 3, 7), (2, 3, 8)</td><td></td></tr><tr><td>29, 31, 37, 43</td><td>(2, 3, 7)</td><td></td></tr><tr><td rowspan="3">5</td><td>3</td><td>(2, 2, 2, m<sub>4</sub>)</td><td></td></tr><tr><td>5</td><td>(2, 2, 2, 3), (2, 2, 2, 4)</td><td></td></tr><tr><td>7</td><td>(2, 2, 2, 3)</td><td></td></tr><tr><td>5</td><td>3</td><td>(2, 2, 2, 2, 2)</td><td></td></tr></tbody></table>
<table><caption>**Tab. B.5.** Hitchin components parametrized by a single non-vanishing differential.</caption><thead><tr><th>Group</th><th>d<sub>N</sub> + 1</th><th>(m<sub>1</sub>, ..., m<sub>k</sub>) with<br>m<sub>i</sub> ≤ m<sub>i+1</sub>, ∑<sub>m<sub>i</sub></sub> 1/m<sub>i</sub> &lt; k-2</th></tr></thead><tbody><tr><td rowspan="2">PGL(3, ℝ)</td><td>2</td><td>(2, 2, 2, 2, 2) or (2, 2, 2, m<sub>4</sub>)</td></tr><tr><td>3</td><td>(m<sub>1</sub>, m<sub>2</sub>, m<sub>3</sub>) with m<sub>1</sub> ≥ 3</td></tr><tr><td rowspan="2">PGL(4, ℝ)</td><td>3</td><td>(3, 3, m<sub>3</sub>)</td></tr><tr><td>4</td><td>(2, m<sub>2</sub>, m<sub>3</sub>) with m<sub>2</sub> ≥ 4</td></tr><tr><td rowspan="2">PGL(5, ℝ)</td><td>3</td><td>(3, 3, 4)</td></tr><tr><td>4</td><td>(2, 4, m<sub>3</sub>)</td></tr><tr><td rowspan="2">PGL(n, ℝ) with n = 6, 7</td><td>4</td><td>(2, 4, 5)</td></tr><tr><td>6</td><td>(2, 3, m<sub>3</sub>)</td></tr><tr><td>PGL(n, ℝ) with n = 8, 9, 10, 11</td><td>6</td><td>(2, 3, 7)</td></tr><tr><td>PSp<sup>±</sup>(4, ℝ) or PO(2, 3)</td><td>4</td><td>(m<sub>1</sub>, m<sub>2</sub>, m<sub>3</sub>) with m<sub>2</sub> ≠ 3</td></tr><tr><td rowspan="2">PSp<sup>±</sup>(6, ℝ) or PO(3, 4)</td><td>4</td><td>(2, 4, 5) or (2, 5, 5)</td></tr><tr><td>6</td><td>(2, 3, m<sub>3</sub>) or (3, 3, m<sub>3</sub>)</td></tr><tr><td>PSp<sup>±</sup>(2m, ℝ) or PO(m, m+1) with m = 4, 5</td><td>6</td><td>(2, 3, 7)</td></tr><tr><td>PO<sup>±</sup>(4, 4)</td><td>6</td><td>(2, 3, m<sub>3</sub>) or (3, 3, m<sub>3</sub>)</td></tr><tr><td>PO<sup>±</sup>(5, 5)</td><td>6</td><td>(2, 3, 7)</td></tr><tr><td>G<sub>2</sub></td><td>6</td><td>(m<sub>1</sub>, m<sub>2</sub>, m<sub>3</sub>) with (m<sub>1</sub>, m<sub>3</sub>) ≠ (2, 5)</td></tr></tbody></table>

*Acknowledgments.* We thank Brian Collier, Olivier Guichard, Qiongling Li, Anna Wienhard and Tengren Zhang for helpful conversations, and the referee for various suggestions that have helped improve the exposition of the material presented in this paper.

*Funding.* D. Alessandrini was supported by the DFG grant AL 1978/1-1 within the Priority Programme SPP 2026 “Geometry at Infinity”. G.-S. Lee was supported by the DFG research grant “Higher Teichmüller Theory”, by the European Research Council under ERC-Consolidator Grant 614733, and by the DFG grant LE 3901/1-1 within the Priority Programme SPP 2026 “Geometry at Infinity”. F. Schaffhauser was supported by Convocatoria 2018-2019 de la Facultad de Ciencias (Uniandes), Programa de investigación “Geometría y Topología de los Espacios de Módulos”, the European Union’s Horizon 2020 research and innovation programme under grant agreement No 795222 and the University of Strasbourg Institute of Advanced Study (USIAS). The authors acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).

## References

[1] Alling, N. L., Greenleaf, N.: Foundations of the Theory of Klein Surfaces. Lecture Notes in Math. 219, Springer, Berlin (1971) Zbl 0225.30001 MR 0333163

[2] Baraglia, D.: Cyclic Higgs bundles and the affine Toda equations. Geom. Dedicata 174, 25–42 (2015) Zbl 1321.53027 MR 3303039

[3] Bridgeman, M., Canary, R., Labourie, F., Sambarino, A.: The pressure metric for Anosov representations. Geom. Funct. Anal. 25, 1089–1179 (2015) Zbl 1360.37078 MR 3385630

[4] Bridgeman, M., Canary, R., Labourie, F., Sambarino, A.: Simple root flows for Hitchin representations. Geom. Dedicata 192, 57–86 (2018) Zbl 1383.53038 MR 3749423

[5] Brin, M. G.: Seifert fibered spaces. Notes for a course given in the Spring of 1993. arXiv:0711.1346v2 (2007)
[6] Burelle, J.-P.: Rigidity of diagonally embedded triangle groups. Canad. Math. Bull. **64**, 560–579 (2021) Zbl **07400598** MR **4313550**

[7] Burger, M., Iozzi, A., Parreau, A., Pozzetti, B.: Currents, systoles and compactifications of character varieties. Proc. London Math. Soc. (3) **123**, 565–596 (2021) MR **4368682**

[8] Choi, S., Goldman, W. M.: Convex real projective structures on closed surfaces are closed. Proc. Amer. Math. Soc. **118**, 657–661 (1993) Zbl **0810.57005** MR **1145415**

[9] Choi, S., Goldman, W. M.: The deformation spaces of convex $\mathbb{R}P^2$-structures on 2-orbifolds. Amer. J. Math. **127**, 1019–1102 (2005) Zbl **1086.57015** MR **2170138**

[10] Choi, S., Lee, G.-S.: Projective deformations of weakly orderable hyperbolic Coxeter orbifolds. Geom. Topol. **19**, 1777–1828 (2015) Zbl **1333.57028** MR **3375519**

[11] Collier, B.: Finite order automorphisms of Higgs bundles: Theory and application. PhD thesis, Univ. of Illinois at Urbana-Champaign (2016) MR **3624308**

[12] Collier, B., Li, Q.: Asymptotics of Higgs bundles in the Hitchin component. Adv. Math. **307**, 488–558 (2017) Zbl **1372.30046** MR **3590524**

[13] Cooper, D., Long, D., Thistlethwaite, M.: Computing varieties of representations of hyperbolic 3-manifolds into $SL(4, \mathbb{R})$. Experiment. Math. **15**, 291–305 (2006) Zbl **1117.57016** MR **2264468**

[14] Cooper, D., Long, D. D., Thistlethwaite, M. B.: Flexing closed hyperbolic manifolds. Geom. Topol. **11**, 2413–2440 (2007) Zbl **1142.57009** MR **2372851**

[15] Corlette, K.: Flat $G$-bundles with canonical metrics. J. Differential Geom. **28**, 361–382 (1988) Zbl **0676.58007** MR **965220**

[16] Dai, S., Li, Q.: Minimal surfaces for Hitchin representations. J. Differential Geom. **112**, 47–77 (2019) Zbl **07054919** MR **3948227**

[17] Dai, S., Li, Q.: On cyclic Higgs bundles. Math. Ann. **376**, 1225–1260 (2020) Zbl **1448.14030** MR **4081114**

[18] Damianou, P. A.: A beautiful sine formula. Amer. Math. Monthly **121**, 120–135 (2014) Zbl **1307.17016** MR **3149030**

[19] Donaldson, S. K.: Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3) **55**, 127–131 (1987) Zbl **0634.53046** MR **887285**

[20] Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Math. Ser. 49, Princeton Univ. Press, Princeton, NJ (2012) Zbl **1245.57002** MR **2850125**

[21] Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. **103**, 1–211 (2006) Zbl **1099.14025** MR **2233852**

[22] García-Prada, O., Peón-Nieto, A., Ramanan, S.: Higgs bundles for real groups and the Hitchin–Kostant–Rallis section. Trans. Amer. Math. Soc. **370**, 2907–2953 (2018) Zbl **1395.14029** MR **3748589**

[23] García-Prada, O., Wilkin, G.: Action of the mapping class group on character varieties and Higgs bundles. Doc. Math. **25**, 841–868 (2020) Zbl **1451.14108** MR **4137937**

[24] Goldman, W. M.: Convex real projective structures on compact surfaces. J. Differential Geom. **31**, 791–845 (1990) Zbl **0711.53033** MR **1053346**

[25] Gorbatsevich, V. V., Onishchik, A. L.: Lie transformation groups. In: Lie Groups and Lie Algebras, I, Encyclopaedia Math. Sci. 20, Springer, Berlin, 95–235 (1993) Zbl **0797.22001** MR **1306739**

[26] Guéritaud, F., Guichard, O., Kassel, F., Wienhard, A.: Anosov representations and proper actions. Geom. Topol. **21**, 485–584 (2017) Zbl **1373.37095** MR **3608719**

[27] Guichard, O.: Composantes de Hitchin et représentations hyperconvexes de groupes de surface. J. Differential Geom. **80**, 391–431 (2008) Zbl **1223.57015** MR **2472478**

[28] Guichard, O., Wienhard, A.: Convex foliated projective structures and the Hitchin component for $\mathrm{PSL}_4(\mathbb{R})$. Duke Math. J. **144**, 381–445 (2008) Zbl **1148.57027** MR **2444302**
[29] Guichard, O., Wienhard, A.: Anosov representations: domains of discontinuity and applications. Invent. Math. **190**, 357–438 (2012) Zbl **1270.20049** MR **2981818**

[30] Guichard, O.: Zariski closure of positive and maximal representations. In preparation

[31] Heller, S., Schaposnik, L. P.: Branes through finite group actions. J. Geom. Phys. **129**, 279–293 (2018) Zbl **1388.81546** MR **3789253**

[32] Heusener, M., Porti, J.: Infinitesimal projective rigidity under Dehn filling. Geom. Topol. **15**, 2017–2071 (2011) Zbl **1237.57016** MR **2860986**

[33] Hitchin, N.: Stable bundles and integrable systems. Duke Math. J. **54**, 91–114 (1987) Zbl **0627.14024** MR **885778**

[34] Hitchin, N. J.: Lie groups and Teichmüller space. Topology **31**, 449–473 (1992) Zbl **0769.32008** MR **1174252**

[35] Ho, N.-K., Liu, C.-C. M.: Yang–Mills connections on orientable and nonorientable surfaces. Mem. Amer. Math. Soc. **202**, no. 948, viii+98 pp. (2009) Zbl **1191.53021** MR **2561624**

[36] Ho, N.-K., Wilkin, G., Wu, S.: Hitchin's equations on a nonorientable manifold. Comm. Anal. Geom. **26**, 857–886 (2018) Zbl **1397.53095** MR **3853929**

[37] Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects. Lect. Math. ETH Zürich, Birkhäuser, Basel (1997) Zbl **0896.53002** MR **1451625**

[38] Kapovich, I., Benakli, N.: Boundaries of hyperbolic groups. In: Combinatorial and Geometric Group Theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296, Amer. Math. Soc., Providence, RI, 39–93 (2002) Zbl **1044.20028** MR **1921706**

[39] Kapovich, M., Leeb, B., Porti, J.: Dynamics on flag manifolds: domains of proper discontinuity and cocompactness. Geom. Topol. **22**, 157–234 (2018) Zbl **1381.53090** MR **3720343**

[40] Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. **81**, 973–1032 (1959) Zbl **0099.25603** MR **114875**

[41] Kostant, B.: Lie group representations on polynomial rings. Amer. J. Math. **85**, 327–404 (1963) Zbl **0124.26802** MR **158024**

[42] Kostant, B., Rallis, S.: Orbits and representations associated with symmetric spaces. Amer. J. Math. **93**, 753–809 (1971) Zbl **0224.22013** MR **311837**

[43] Labourie, F.: Anosov flows, surface groups and curves in projective space. Invent. Math. **165**, 51–114 (2006) Zbl **1103.32007** MR **2221137**

[44] Labourie, F.: Cross ratios, Anosov representations and the energy functional on Teichmüller space. Ann. Sci. École Norm. Sup. (4) **41**, 437–469 (2008) Zbl **1160.37021** MR **2482204**

[45] Labourie, F.: Cyclic surfaces and Hitchin components in rank 2. Ann. of Math. (2) **185**, 1–58 (2017) Zbl **1372.32021** MR **3583351**

[46] Labourie, F., McShane, G.: Cross ratios and identities for higher Teichmüller–Thurston theory. Duke Math. J. **149**, 279–345 (2009) Zbl **1182.30075** MR **2541705**

[47] Larsen, M., Lubotzky, A.: Representation varieties of Fuchsian groups. In: From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math. 28, Springer, New York, 375–397 (2013) Zbl **1272.20054** MR **2986968**

[48] Lewittes, J.: Automorphisms of compact Riemann surfaces. Amer. J. Math. **85**, 734–752 (1963) Zbl **0146.10403** MR **160893**

[49] Li, Q.: Teichmüller space is totally geodesic in Goldman space. Asian J. Math. **20**, 21–46 (2016) Zbl **1338.57019** MR **3460757**

[50] Long, D. D., Reid, A. W.: Commensurability and the character variety. Math. Res. Lett. **6**, 581–591 (1999) Zbl **0962.57006** MR **1739217**

[51] Long, D. D., Thistlethwaite, M. B.: The dimension of the Hitchin component for triangle groups. Geom. Dedicata **200**, 363–370 (2019) Zbl **1415.22010** MR **3956201**

[52] Marquis, L.: Espace des modules de certains polyèdres projectifs miroirs. Geom. Dedicata **147**, 47–86 (2010) Zbl **1243.52015** MR **2660566**
[53] Schaffhauser, F.: Lectures on Klein surfaces and their fundamental group. In: Geometry and Quantization of Moduli Spaces, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 67–108 (2016) Zbl 1388.30053 MR 3675463

[54] Schaffhauser, F.: On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles. J. Differential Geom. **105**, 119–162 (2017) Zbl 1360.30037 MR 3592696

[55] Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. **15**, 401–487 (1983) Zbl 0561.57001 MR 705527

[56] Serre, J.-P.: Sur la semi-simplicité des produits tensoriels de représentations de groupes. Invent. Math. **116**, 513–530 (1994) Zbl 0816.20014 MR 1253203

[57] Serre, J.-P.: Complète réductibilité. In: Séminaire Bourbaki, Vol. 2003/2004, exp. 932, Astérisque **299**, viii, 195–217 (2005) Zbl 1156.20313 MR 2167207

[58] Simpson, C. T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Amer. Math. Soc. **1**, 867–918 (1988) Zbl 0669.58008 MR 944577

[59] Simpson, C. T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. **75** 5–95 (1992) Zbl 0814.32003 MR 1179076

[60] Simpson, C.: Katz’s middle convolution algorithm. Pure Appl. Math. Quart. **5**, 781–852 (2009) Zbl 1176.14002 MR 2508903

[61] Thurston, W. P.: Three-dimensional Geometry and Topology. Vol. 1. Princeton Math. Ser. 35, Princeton Univ. Press, Princeton, NJ (1997) Zbl 0873.57001 MR 1435975

[62] Thurston, W. P.: Geometry and Topology of 3-Manifolds. Lecture notes, Princeton Univ. (1979)

[63] Weir, E. A.: The dimension of the restricted Hitchin component for triangle groups. PhD thesis, Univ. of Tennessee (2018)

[64] Wienhard, A.: An invitation to higher Teichmüller theory. In: Proc. International Congress of Mathematicians (Rio de Janeiro, 2018), Vol. II, World Sci., Hackensack, NJ, 1013–1039 (2018) Zbl 1447.32023 MR 3966798