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I've ran into the term "Higher Genus Hyperbolic Surface" in the context of Ricci flows of orientable surfaces of dimension $2$.

I'm not familiar with the term, and wasn't able to find a definition in Wikipedia or in any of the books on my desk.

I imagine the definition would be something like:

Connected sum of $g$ tori, endowed with a metric of sectional curvature $-1$.

Is this a correct definition of such a surface? Is there a nice way to visualize it?

Edit: More context.

I've ran into the term mentioned above while reading a discussion called "The Topology and Geometry of Low-Dimensional Manifolds" in P. Topping's "Lectures on the Ricci Flow".

The discussion aims to motivate the development of the Ricci Flow, as a tool for classifying such manifolds. It's informal in flavor. We assume all manifolds are compact and orientable. See bellow for summary.

The writer mentions that 2-dimensional manifolds are classified by the genus. After discussing that fact we're told:

It turns out each of such surface can be endowed with conformally equivalent metric of constant Gaussian curvature ... the universal cover of the surface must be $S^2,\mathbb{R}^2,\mathbb{H}^2$, and the original surface is than described as quotient of it's universal cover by a group of isometries acting freely.

This gives rise to $S^2$, a flat torus, or a higher genus hyperbolic surface, depending if the curvature is $1,0$ or $-1$ (up to uniform scaling of the metric).

NG_
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  • I guess this is a surface of genus $>1$, endowed with an hyperbolic structure. This may be related. – TheSilverDoe May 05 '23 at 11:10
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    Out of context this question is impossible to answer. Are we to understand the phrase "Higher Genus Hyperbolic Surface" in contrast to "Lower Genus Hyperbolic Surface"? – Lee Mosher May 05 '23 at 20:29
  • I've added some context @LeeMosher. Feedback appreciated. – NG_ May 06 '23 at 11:22
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    In this case, the "higher genus hyperbolic surface" is in contrast to "lower genus non-hyperbolic surface". – Mr. Brown May 06 '23 at 12:45
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    A genus $g$ surface with constant curvature can be visualized as a $4g$-sided polygon, with the same shape as in the regular hyperbolic tiling ${4g,4g}$, with opposite sides attached. This is a generalization of the torus ($g=1$) being a square with opposite sides attached. – mr_e_man May 17 '23 at 22:01
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    For example, the genus $2$ surface is the regular octagon with $45^\circ$ vertex angles, as in https://en.wikipedia.org/wiki/Order-8_octagonal_tiling – mr_e_man May 17 '23 at 22:06

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From the quote you gave, it looks like "higher genus" simply means "genus higher than $1$", as suggested by @TheSilverDoe. Your description of such a surface is therefore appropriate if you add the requirement $g > 1$.

Learning to visualize such surfaces and their hyperbolic structures is a big, big topic. A good starting point is Jeff Week's book The Shape of Space. You can also pick up any textbook on hyperbolic manifolds.

Lee Mosher
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