一般有限表现群的小消去稳定性与同构刚性
Small Cancellation Stability and Isomorphism Rigidity for Generic Finitely Presented Groups
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中文总结 AI 辅助
该研究证明一般有限表现群的同构刚性，给出一般情形下求解m生成q关系群同构问题的二次时间算法，并确定对应同构类型数量的渐近表达式。
中文摘要 AI 辅助
设 $F_m=F(a_1,\dots,a_m)$ 满足 $m\ge2$，并固定 $q\ge1$。对每个固定的 $0<\lambda<1$，我们证明：长度为 $n$ 的独立均匀随机循环约化字构成的 $q$ 元组 $\mathbf W_n$ 是 $\lambda$-稳定的，其概率以指数速度收敛到1。具体而言，对每个 $\Phi\in Aut(F_m)$，元组 $\Phi(\mathbf W_n)$ 经循环约化和对称化后满足 $C'(\lambda)$ 小消去条件。结合一般 $\lambda$-稳定性、Greendlinger 正规闭包刚性，以及 Kapovich-Schupp-Shpilrain 关于一般 Nielsen 唯一性和一般 Whitehead 刚性的已有结果，我们对任意固定的 $m\ge2,q\ge1$，建立了一般 $m$ 生成 $q$ 关系群的同构刚性。由此证明：两个此类一般群 $\langle a_1,\dots,a_m| r_1,\dots, r_q\rangle$ 和 $\langle a_1,\dots, a_m| s_1,\dots, s_q\rangle$ 同构当且仅当，在可能对生成元 $a_1,\dots,a_m$ 进行置换和取逆后，关系元组 $(r_1,\dots, r_q)$ 与 $(s_1,\dots, s_q)$ 可通过关系的重排、循环置换和取逆完全一致。作为应用，我们得到一个二次时间算法，可在一般情形下求解 $m$ 生成 $q$ 关系群的同构问题，并证明：由长度为 $n$ 的循环约化关系构成的 $m$ 生成 $q$ 关系表示所对应的同构类型数量，渐近为 $\frac{(2m-1)^{qn}}{2^{m+q}m!\\,q!\\,n^q}$。一般 $\lambda$-稳定性的证明依赖于测地流的应用，以及我们提出的确定性充分准则：$F_m$ 中的 $q$ 元组 $\mathbb W$ 是 $\lambda$-稳定的，当且仅当 $\mathbb W$ 的各分量充分投影逼近填充流。
英文摘要
Let $F_m=F(a_1,\dots,a_m)$ with $m\ge 2$, and fix $q\ge 1$. For every fixed $0<λ<1$, we prove that a $q$-tuple $\mathbf W_n$ of independent uniformly random cyclically reduced words of length $n$ is \emph{$λ$-stable} with probability converging to $1$ exponentially fast. Namely, for every $Φ\in Aut(F_m)$, the tuple $Φ(\mathbf W_n)$, after cyclic reduction and symmetrization, satisfies the $C'(λ)$ small cancellation condition. Combining generic $λ$-stability with Greendlinger normal-closure rigidity and with previous results of Kapovich-Schupp-Shpilrain on generic Nielsen uniqueness and generic Whitehead rigidity we establish, for any fixed $m\ge 2, q\ge 1$, isomorphism rigidity for generic $m$-generator $q$-relator groups. Thus we show that two such generic groups $\langle a_1,\dots, a_m| r_1,\dots, r_q\rangle$ and $\langle a_1,\dots, a_m| s_1,\dots, s_q\rangle$ are isomorphic if and only if, after possibly permuting and inverting the generators $a_1,\dots, a_m$, the relator tuples $(r_1,\dots, r_q)$ and $(s_1,\dots, s_q)$ are the same, up to reordering, cyclic permutations and inverting the relators. Among the applications, we obtain a quadratic-time algorithm that generically solves the isomorphism problem for $m$-generator $q$-relator groups, and show that the number of isomorphism types represented by $m$-generator $q$-relator presentations with cyclically reduced relators of length $n$ is asymptotic to \[ \frac{(2m-1)^{qn}}{2^{m+q}m!\,q!\,n^q}. \] The proof of generic $λ$-stability relies on the use of geodesic currents and on our deterministic sufficient criterion for a $q$-tuple $\mathbb W$ in $F_m$ to be $λ$-stable in terms of the components of $\mathbb W$ being sufficiently projectively close to filling currents.
