The methods in that book may fail, though. Check out the one on hypergeometric function proofs with computers. I tried to read *that* book too early - probably accidentally downloaded it when I had first learned in 7th grade what a wonderful place the internet was for "semi-legal" downloads - and settled for gfology instead.
What if I take the Cayley graph of $\Bbb Z$ with generating set $\langle \pm 1, \pm 2\rangle$ instead of $\langle \pm 1 \rangle$? How can you prove that the Gromov growth are asymptotically the same?
It's nontrivial.
All of it goes back to the idea of quasi-isometry.
I'm looking at the Cayley graph as a sort of drawing in the Euclidean plane, never mind that it has its own topology and whatnot. I'm just looking at it as a cool fractal-ish drawing.
Now what is the metric in the Euclidean plane? You know, of course.
Then I can't make sense out of your "what's the distance between 1 and [blah]". If you want to retrict your metric to the graph, then the distance is just obtained from squareing, summing, and square-rooting
@SohamChowdhury $d(x, y) = \sqrt{x^2 + y^2}$. To find the distance with this metric, just embed properly, mark the points, forget about the graph , and draw a straightline between them.
@BalarkaSen My question is this: if you look at the Cayley graph as a drawing on the plane, what is the x-coordinate of the point corresponding to $\lim_{n\rightarrow \infty} (yxyx^{-1})^n$?
For example, let the distance between $1$ and $x$ be $1/2$, the distance between $1$ and $x^2$ be $1/2 + 1/4$, etc. so that the distance between the origin and $\lim_{x\to \infty} x^n = 1$.
@BalarkaSen Anyway, let me properly state the question. What is the x-coordinate of the point corresponding to $\lim_{n\rightarrow \infty} (yxyx^{-1})^n$, when I define the embedding (?) of $\Gamma(F_2)$ into $\Bbb{R^2}$in this manner?
Don't have to think about it, just wanted to make sure my question was well-defined. :P
$Int(A∪B)⊆Int(A)∪Int(B)$: takeA=(0,1] and B=[1,2). Then Int(A)=(0,1), Int(B)=(1,2), but Int(A∪B)=(0,2). Remember this question @Balarka I found an example Is this thing right?
Every question I am doing my intuition tells me that it is obvious I always feel it is obvious But I cant write it into words Never happened to me in other parts of maths
No i am not talking easy Other proofs were damn hard But i was able to write them into words But here it is really difficult to write them into words@Soham
Hey @balarka I thought of a fact Can i use that $C\subseteq D \implies \operatorname{int}C \subseteq \operatorname{int}D $
Because $int(A \cup B)$ is basically the union of all open subsets of $A \cup B$ and the union will have at least one set which will be in $ int(A) \cup int(B)$ right @Balarka
Can you solve this problem? "$n$ children sit in a line. If they are allowed to move from their positions by at most one place, in how many ways can they sit?" (Hard version: replace "line" with "circle".)
It's very similar to this problem: find the number of disjoint (ordered) pairs of subsets of $\{1\cdots n\}$. Apparently it's an IIT-JEE problem, but I found it in Engel first.
Well @Balarka Is there any such set such that if i take any point in the set and take a ball of arbitrary radius around the point the ball will always lie inside the set?
