至多差有界误差的相同
第 2 篇留给我们一个尴尬：换生成元时字度量会变。补救之道是宣布：若两个度量空间在忽略有界误差后一致，就视为「相同」。映射 f: X → Y 是拟等距，若存在常数 L ≥ 1，C ≥ 0，使对一切 x, x′ 有 (1/L)·d(x,x′) − C ≤ d(f(x), f(x′)) ≤ L·d(x,x′) + C，且 Y 中每点都在像的 C 邻域内。它无需连续或单射——这是从极远处看到的几何。
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.群增长得多快
你能算的第一个拟等距不变量。增长函数 β(n) 数到单位元距离 n 内的元素个数：β(n) = #{g : |g| ≤ n}，即 n-球的大小。它的增长率——多项式、指数、或介乎其间——是拟等距不变量（在函数的自然等价意义下）。它侦测深层结构：当群含有足够的自由性时呈指数增长，当群近乎阿贝尔时呈多项式增长。
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.