INTRODUCTION The decays of the τ lepton into three pseudoscalar particles can provide information on hadronic form factors, K ∗ resonance spectroscopy, and the Wess-Zumino anomaly [1], and also can be used for studies of CP violation in the leptonic sector [2]. By studying decays into final states that contain one or three kaons, it is possible to extract strange spectral functions, which can be used for a direct determination of the strange quark mass and the Cabibbo-Kobayashi-Maskawa (CKM) matrix element | V$_{us}$ | [3–5]. These three hadron decays have been studied since the initial discovery of the τ lepton, but only the decay to three charged pions has been closely investigated. Even for this mode, measurements of its branching fraction have only been provided by the CLEO [6] and BaBar [7] groups. For all other modes, there are still many unresolved problems. For example, the branching fraction for τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ decay recently measured by the BaBar collaboration [7] is significantly lower than the values from most previous measurements [8]; there is large scatter in the measured central values, which is reflected in the large scale factor (2.1) applied by the Particle Data Group when forming their average [8]. Intermediate two- and three-body resonance states in this decay have been previously studied but only with limited statistical precision, see, e.g., Refs [9–11]. There is evidence for quite rich dynamics, with strong signals for the K$_{1}$ (1270), K$_{1}$ (1400) and K $^{∗}$(1410) resonances in the three-body final state; other resonances can be seen in various two-body sub-systems. Theoretically, two intermediate resonances, the ρ (770) 0 and K $^{∗}$(892) 0 (or its excitations), are expected to contribute to τ decay into K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ : τ − → K $^{−}$ρ (770) $^{0}$( → π $^{+}$π $^{−}$) ν$_{τ}$ and τ − → K $^{∗}$(892) $^{0}$( → K $^{−}$π $^{+}$) π $^{−}$ν$_{τ}$ [12]. These are shown in Figs 1(a) and (b), respectively. Here we present new measurements of the branching fractions for τ − → π $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$π $^{+}$π $^{−}$ν$_{τ}$ , τ − → K $^{−}$K $^{+}$π $^{−}$ν$_{τ}$ , and τ − → K $^{−}$K $^{+}$K $^{−}$ν$_{τ}$ decays. (Unless otherwise specified, the charge-conjugate decay is also implied throughout this paper.) Events in which a π $^{+}$π − pair is consistent with a K 0 S decay are excluded. Because of particle misidentifica- tion, measurements of all four of the above modes are correlated and, thus, are considered simultaneously. We use a data sample of 666 fb − 1 collected with the Belle detector at the KEKB asymmetric-energy e $^{+}$e − collider [13] on the Υ(4S) resonance, 10.58 GeV, and 60 MeV below it (off-resonance). This sample contains 6 . 12 × 10 8 produced τ -pairs. The Belle de- 4