By Gilman R.H., Hermiller S., Holt D.F.
We end up finitely generated workforce G is nearly loose if and provided that there exists a producing set for G and к > zero such that every one k-locally geodesic phrases with admire to that producing set are geodesic.
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Extra info for A characterisation of virtually free groups
The torus Our interest will be in the cocycle ~*. of the group and some basic properties and the group then a continuous tations L2 represen- SL(2). 33) ~(gl)~ g2 ) = ~(gl, g2)~(gl, g2) for all gl, g2 c G. T (by the associative on which G with values Note that cross-section ~ f 9 Here in ~ is a Borel f u n c t i o n from law in G ) G • G to must be a two-cocycle T . above obviously However, if f' depends on the choice of is another such cross-section, 29 the cocycle it determines will be c o h o m o l o g o u s to projective Thus each r e p r e s e n t a t i o n u n i q u e l y determines an element of A n y Borel map f r o m is called a m u l t i p l i e r simply, a .
1 also generalizes in GL2(R). Z0 and to hlassical to arbitrary number fields. 11) must be replaced by the identity G,~ = with on component k/2 . 1, one must appeal to the fact that (3 11) where that Proposition aj ~G~. G K~ j = l % g c~U Consequently one starts with fj of weight k/2 and defines fj and each piece G-ai GUK~o " trivial on and satisfying correspond Z0 ~ to collections ~ on ~ functions ~ forms (3-7) of Proposition forms of weight situation in the case of totally imaginary fields. 1 now k/2.
Into ~i and is an ~ U(H), whence The rest of the proposition follows from the fact that a n operator e-representations ~' = t~,(g). is a Borel homomorphism of continuous, (b), suppose A intertwines the if and only if it intertwines the corresponding ordinary representations ~I and ~2" We shall now describe Weil's construction in earnest. ian groups G so the notation denote the value of the character at x e G. 19. G is we shall assume throughout that this is the case. Let denote a local field of characteristic and V a finite-dimensional non-degenerate will vector space defined over F.
A characterisation of virtually free groups by Gilman R.H., Hermiller S., Holt D.F.