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What are the ten most fundamental topics in geometric group theory? This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expecte …

Here is a version of the table of contents of Bowditch's lecture notes A course on geometric group theory. Group presentations, free groups, abelianisation. Cayley graphs. Quasi-isometries and thei …

Only for $m=2$, $\sum_{i=1}^mx^{2n+1}_k$ is always divisible by $\sum_{i=1}^mx_k$. Indeed, $$x+y | x^{2n+1}+y^{2n+1}, n \in \mathbb N.$$

I'd like to get a list of instances in mathematics where a problem with two parameters (or some parameter set to $2$) is qualitatively different from the instance of that problem with the value set to …

I don't have 10 things for your list, but I can describe the syllabus of the Introduction to Geometric Group Theory Masters' course I have taught for the last couple of years. I hope this is worth men …

Growing Your Balls A paper was presented at the FOCS '10 conference with title How to Grow Your Balls, see also the comments from the blog linked below; a tutorial was also given, with the more subtle …

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of topo …

Two that I like can be found on p. 756 of Edgar R. Lorch's Amer. Math. Monthly paper "Continuity and Baire functions" (Volume 78, 1971, pp. 748-762): [...] the reader is reminded of the fact that set …

Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher categorie …

The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection of tricks u …

A trick/technique that I like (and used) a lot is the formal geometry approach (after Gelfand-Kazhdan) for passing from a local to a global result. Let $X$ be a $d$-dimensional manifold. There is a an …

One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone seque …

Here is my take. Unlike Andy, I would not structure such a course around big theorems. In part, this is because your students simply do not have enough background to handle any "big theorems." Instead …

This question is seven years old, but since I wanted to ask the same question and provide those examples for induction on prime numbers, I found this question and saw that the first example is missing …

Geometric group theory is a huge subject, and a course that really tried to cover all of it would be too disjointed to be useful. If I were teaching such a course, I would choose a few major theorems …