What kind of definitions need be introduced in a PhD thesis? For all the terminology in mathematics, I am not sure where to draw a line between those which need to be introduced in a PhD thesis and those which do not. For example, I suppose that in a PhD thesis, no one will bother to give the definition of group, or normal subgroup, etc. but probably commutators as it can be defined in two different ways, namely $x^{-1}y^{-1}xy$ and $xyx^{-1}y^{-1}$. For braid group one probably will write down the definition anyway, even though it is well known among topologists. I am not sure about some terminologies such as the lower central series: does it need to be introduced?
Where do we draw the line?
 A: One of the professors I was working with during my PhD and that has the reputation to write deep articles in a very accessible way once gave me that advice: 
"A PhD thesis is not an article. This is one of the few occasions in your career where you are given plenty of space to write everything you want to say. Take that space to add many examples and remarks. Also, a thesis, in addition to the presentation of new important results, is an exposition of the current state of arts of your domain. Explain all definitions and concepts clearly. Think about you at the beginning of your PhD, when you were struggling to understand articles, decyphering what is "clear" or "well-known". You wished you had a place where you could find a clear exposure of these facts. Your thesis should be that place".
I hope it helps.
A: As with any piece of mathematical writing, include definitions if you think it is reasonable that some of your readers will not already know them. Additionally, if you think that your readers will know definitions, but you are aware of multiple conventions, do your readers the kindness of stating which convention you use. If you are unaware of multiple conventions, I suggest not worrying about it.
A: A definition can be also used to introduce a consistent notation. 
For instance, if the thesis deals with different concepts of differentiation, then a definition can be used to fix the notations, like, e.g. $f'(x;h)$ is the directional derivative of $f$ at $x$ in direction $h$, $\partial_{x_i}f(x)$ is a partial derivative.
