By David Fisher

During this paper we produce an invariant for any ergodic, finite entropy motion of a lattice in an easy Lie crew on a finite degree house. The invariant is basically an equivalence type of measurable quotients of a undeniable sort. The quotients are primarily double coset areas and are made out of a Lie workforce, a compact subgroup of the Lie staff, and a commensurability classification of lattices within the Lie workforce.

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**Additional resources for A canonical arithmetic quotient for actions of lattices in simple groups**

**Example text**

X ~ A, define CeX) = {{H n X I H E C(r)} {HnXIHEC(A)} if X if X = (It I H = { {!!.. n ! 4) C(X)}. c(r)} C(An if X if X H-

B !. lay" ::;Cnye '. tA\ ) A 1 3: 0::::' « a. 4 How to determine the overgroups of members of C if7117r(G)1 if 17r(G)1 = Tn and Tn Our results concerning maximality apply only when n = dim(V) is at least 13, so we . adopt the hypothesis n ~ 13 in this section. The other notation carries over from before . 1 6. L 14 (9). Thus if G is for instance the (unique) group which satisfies G = (¢2,5 7 ), th,en G fixes [Hd and has two orbits of size three on t~e remaining six classes. 2) and the HG,i are the corresponding G-associates.

5). r, so that case L holds with n ~ 3. Here Hx = Nx(Hr)' and so Assume now that X 1:. ii occurs, it is cl{;r ~at Hx nIT 1i. ~ = H o' as desired. 6) ifG~r if G of type GL1(q) I 5 n in C2 and which this splits under the action of IT. This number will usually be called c, and thus we obtain a homomorphism from A to Sc, the symmetric group of degree c. Obviously IT i r. 6), I I and so H < N (H-) H r . " ::! HX,i . that equality holds _ that is H 1 e ('I' UI\ ever, It IS not 111 general true 1 neec not be the full normalizer of H- l' ' .

### A canonical arithmetic quotient for actions of lattices in simple groups by David Fisher

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