Residual finiteness of hyperbolic groups

Can every genuine difference between two transformations be noticed in some finite algebraic model? The simplest example is walking along the integers. A move of three steps disappears if positions are recorded modulo three, but remains a move of three when recorded modulo four. The model must respect composition: doing two moves in succession must have the same result as adding their images. The question covers every finitely generated hyperbolic group, whose shortest-path network has uniformly thin triangles. For each group element other than doing nothing, some finite group must detect it; the detector may change with the element.[1]

Two elements of an infinite group can agree in one finite quotient and disagree in another. Must some finite quotient detect each nonidentity element of every word-hyperbolic group? In $\mathbb Z$, a nonzero integer $k$ survives reduction modulo $|k|+1$. In general, require a finite group $F$ and a homomorphism $\varphi:G\to F$ with $\varphi(g)\ne1$ for each $g\ne1$. Equivalently, $\bigcap_{N\trianglelefteq G,\,[G:N]<\infty}N=\{1\}$. The kernels may depend on $g$; this is not a single finite embedding. Word-hyperbolicity means uniformly thin geodesic triangles in a finite-generator Cayley graph, not an assumed faithful matrix representation or cubical action.[1]

For the integer-step group, every nonzero step has a detector: record positions modulo a number larger than its absolute size. This does not give detectors for every hyperbolic group. The cited splitting algorithm still assumes residual finiteness of its specified subgroup, and the 2026 published Introduction explicitly retains the general question. A proof for the full class or a verified hyperbolic counterexample is not established in the selected sources.[1] [2]

The modular calculation verifies the infinite cyclic case, but no comparable argument is established for arbitrary thin-triangle groups in these sources. Corollary 4.8 of the splitting paper retains the quasiconvex subgroup’s residual-finiteness assumption, and the discussion following it makes the finite-quotient question explicit. The 2026 automorphism paper likewise retains Gromov’s question. Conditional residual-finiteness transfers for Artin groups use specified hyperbolic quotients rather than proving the property for all hyperbolic groups.[1] [2] [3]

A finite detector can certify that two transformations are different: apply it to the first followed by the inverse of the second. Such certificates let finite algebra reveal information about an infinite object, although failure to find one is not a certificate of equality. The property also enters structural results. The cited splitting algorithm assumes it for a specified subgroup, and the Artin-group work derives residual finiteness for its stated family only if certain hyperbolic groups have it. A universal answer would therefore affect tools used beyond the original class, under those exact hypotheses.[1] [3]

The intersection of finite-index normal subgroups describes exactly what all finite quotients fail to see. If the intersection is trivial, every unequal pair has a finite separating certificate, making finite-index methods relevant to infinite-group structure. The splitting paper uses such subgroups to control its geometric splitting tests under the stated quasiconvexity and one-endedness assumptions. The Artin-group result illustrates a further transfer: finite detection in selected hyperbolic quotients can support finite detection in a different algebraic family. The connection is conditional and structural, not a promised running-time bound or a claim that the word problem depends on settling this question.[1] [3]