至多差有界誤差的相同
第 2 篇留給我們一個尷尬：換生成元時字度量會變。補救之道是宣布：若兩個度量空間在忽略有界誤差後一致，就視為「相同」。映射 f: X → Y 是擬等距，若存在常數 L ≥ 1，C ≥ 0，使對一切 x, x′ 有 (1/L)·d(x,x′) − C ≤ d(f(x), f(x′)) ≤ L·d(x,x′) + C，且 Y 中每點都在像的 C 鄰域內。它無需連續或單射——這是從極遠處看到的幾何。
Why generators don't matter (up to QI):
Let S, T be two finite generating sets of G.
Each t in T is a word in S, so |t|_S <= M := max_{t in T} |t|_S.
Then for any g: |g|_S <= M * |g|_T (replace each T-letter by <= M S-letters).
Symmetrically |g|_T <= M' * |g|_S.
Hence (1/M') |g|_T <= |g|_S <= M |g|_T : identity map (G,d_S) -> (G,d_T)
is a quasi-isometry with L = max(M, M'), C = 0.
Consequence: a finitely generated group has a WELL-DEFINED
quasi-isometry type, independent of generators. Now QI invariants
are honest invariants OF THE GROUP.群增長得多快
你能算的第一個擬等距不變量。增長函數 β(n) 數到單位元距離 n 內的元素個數：β(n) = #{g : |g| ≤ n}，即 n-球的大小。它的增長率——多項式、指數、或介乎其間——是擬等距不變量（在函數的自然等價意義下）。它偵測深層結構：當群含有足夠的自由性時呈指數增長，當群近乎阿貝爾時呈多項式增長。
Counting balls:
Z : |k| = |k|, so ball of radius n has 2n+1 elements. beta(n) ~ n (LINEAR)
Z^2 : ball is a diamond |x|+|y| <= n. beta(n) = 2n^2+2n+1 ~ n^2 (QUADRATIC)
Z^d : beta(n) ~ n^d (POLYNOMIAL deg d)
F_2=<a,b> : reduced words. #{length exactly n} = 4*3^(n-1) for n>=1.
beta(n) = 1 + 4(3^n - 1)/2 ~ 3^n (EXPONENTIAL)
The free group's tree branches by 3 at every step (you can't immediately
backatrack), so the ball size triples -- exponential growth is visible
in the picture.