Let $S$ be a closed orientable surface of genus at least 2. I'm interested in the connected components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ for $n$ at least 3. I know that for $n$ odd there is only 3 connected components. My first question is:

1. For n even, do we still have 3 connected components?

Suppose n is odd. There is the connected component containing the trivial representation, the Hitchin component and another one. My second question is:

2. What do we know about the connected component which doen't contains the trivial or Hitchin representations? Can we construct an explicit representation in it? Can any representation in it be deformed to have image in a compact group?

Thanks

Let $S$ be a closed orientable surface of genus at least $2$. I'm interested in the connected components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ for $n$ at least $3$. I know that for $n$ odd there is only $3$ connected components. My first question is:

1. For $n$ even, do we still have $3$ connected components?

Suppose $n$ is odd. There is the connected component containing the trivial representation, the Hitchin component and another one. My second question is:

2. What do we know about the connected component which doen't contains the trivial or Hitchin representations? Can we construct an explicit representation in it? Can any representation in it be deformed to have image in a compact group?

Thanks

Let $S$ be a closed orientable surface of genus at least 2. I'm interested in the connected components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ for $n$ at least 3. I know that for $n$ odd there is only 3 connected components. My first question is:

1. For n even do we still have 3 connected components?

Suppose n is odd. There is the connected component containing the trivial representation, the Hitchin component and another one. My second question is:

2. What do we know about the connected component which doen't contains the trivial representation or Hitchin representations? Can we construct an explicit representation in it? Can any representation in it be deformed to have image in a compact group?

Thanks

Let $S$ be a closed orientable surface of genus at least 2. I'm interested in the connected components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ for $n$ at least 3. I know that for $n$ odd there is only 3 connected components. My first question is:

1. For n even, do we still have 3 connected components?

Suppose n is odd. There is the connected component containing the trivial representation, the Hitchin component and another one. My second question is:

2. What do we know about the connected component which doen't contains the trivial or Hitchin representations? Can we construct an explicit representation in it? Can any representation in it be deformed to have image in a compact group?

Thanks