Confusion about an author's use of "$A\backslash B$" to indicate a quotient

Question

I am currently reading Peter Busher's Geometry and Spectra of Compact Riemann Surfaces. There is a theorem saying that $\pi:S \to \Gamma\backslash S$ is a local isometry if $S$ is a hyperbolic surface and $\Gamma\subset \operatorname{Is}^{+}(S)$ has certain properties.

I thought "$A\backslash B$" meant "$A-B$", but the author says it's a quotient. I was wondering if this was the same expression as $S/\Gamma$ s.t. $$\Gamma\backslash S=S/\Gamma=\left\{\{y\in S\mid y=\varphi(x)\}\mid \varphi\in\Gamma\right\}$$

Answer 1

I have two things to say about this. The first is some general advice, the second is a more direct answer to the question.

1. Notation in a text (a book, journal article, chalkboard, whatever) means exactly what the author tells you it means. While there is often conventional notation for certain ideas, and while it is generally inadvisable to use notation other than the conventional notation, there is nothing inherent in mathematics which prevents such uses. And sometimes different conventions dominate in different cultures. For example, do we have $$ \mathbb{N} = \{ 1, 2, 3, \dotsc \} \qquad\text{or}\qquad \mathbb{N} = \{ 0, 1, 2, \dotsc \}?$$

   Adding to this difficulty is the fact that we only have a finite number of symbols (numerals, letters, operators, etc), and a very large number of objects to describe. It is expected that symbols will sometimes become overloaded and used for more than one thing. My favorite example of this is the word "normal". Another good example is $$ (a, b). $$ Does this represent an open interval in $\mathbb{R}$? an ordered pair? the dual pairing of vectors from a Banach space and its dual?

   At the end of the day, an author may define notation to mean whatever they want it to mean, and might eschew conventional notation (or may come from a part of mathematics where the conventions are different from what the reader expects). Hence it is important to pay attention and learn what an author means by a given notation.

2. In parts of mathematics where the actual objects in a set are important (set theory, maybe analysis and PDE, etc), the notation $$ A \setminus B $$ is the collection of points which are in $A$ but not $B$, i.e. the difference set (note: this is typeset with A \setminus B). However, in the context of, for example, Riemannian geometry, it is common to use this notation to represent the quotient by a left group action. That is, $$ \Gamma \backslash S = \{ \Gamma s \mid s \in S \}$$ (note: this is typeset with \Gamma \backslash S). This mirrors the notation for a quotient by a right group action, i.e. $$ S / \Gamma = \{ s\Gamma \mid s \in S \}, $$ but emphasizes the fact that $\Gamma$ acts on the left (but maybe not the right). It might be worth noting that some authors will adjust the notation a little bit to use vertical alignment to clarify the meaning. For example $$ ^{S}/_{\Gamma}, \qquad\text{or}\qquad _{\Gamma}\backslash^{S}. $$ MathJax typesetting is somewhat imperfect, but this can actually look pretty good if typeset by an actual TeX system. There is further discussion of this on the TeX StackExchange.