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

By David Fisher

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

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

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In p a r t i c u l a r ~ vi set of v e r t e x s t a b i l i z e r s u b g r o u p s forms a under does not c o n t a i n vi G G for any v e r t e x path" Gel , - = en ... ,he~gl ,hge~l a source G eI g E Gv0 ,e~n,he~Sn or §8 be an edge first points v, edge to of in the v denoted G[e,v] X. For any v e r t e x geodesic and w r i t e v ÷ e. = {g E G I e ÷ g v ~ , from e + v; le v of to v otherwise, For any v e r t e x X, v if t h e n we e of a n d let us t h e n w r i t e points X let 30 GROUPS ACTING ON GRAPHS G[e] = G[e,le].

32]). i E, and H of a free ~roup (G:H) rank G then - 1) + 1. and let Then the maximal are finite G Y be the bouquet subtree T of Y is 37 THE TRIVIAL just the u n i q u e v e r t e x of CASE Y, and G = n(Y,T) so acts f r e e l y on the tree X = F(Y,T). 1. 8 is (G:H) trivial of [E(H\X) I = (G:H)IE(G\X) I : (G:H)(rank (Bass-Serre, srouP, and H G\X, - G - + acts is finite so if rank freely then G is also 1 (G:H) I V ( G \ X ) I - 1) + 1. + 1 D of C h a p t e r and in fact Serre [77]). i we get a very Let any subgroup of with each c o n j q s a t e p~(v)p (p e z(~,T), ~:Y + z(~,T) be which has of the image of each v E V(Y)).

7 index can graph whose order. generalization (Serre If [77]). (Gi'F) .. 2. THEOREM Proof. is that, group of a c o n n e c t e d are finite of bounded Let us note the f o l l o w i n g formula, and c l e a r l y Since F is i s o m o r p h i c the n u m b e r : subgroup l E(Y) of F\X IF\G/Gel _ subgroup H G. 1. G:r) [ IGel ' E(Y) seen s t a t e m e n t s of of finite V(Y) = for vertices. subgroups and the vertex groups is Now gives the desired result. We have F 1 [Gel to the f u n d a m e n t a l E(Y) and s i m i l a r l y is finite does not meet any conjugate of edges [ Y More G.