By M. Bachir Bekka

ISBN-10: 0521660300

ISBN-13: 9780521660303

The learn of geodesic flows on homogeneous areas is a space of study that has lately yielded a few interesting advancements. This booklet specializes in a lot of those, with one in every of its highlights an undemanding and whole facts through Margulis and Dani of Oppenheim's conjecture. different gains are self-contained remedies of an exposition of Ratner's paintings on Raghunathan's conjectures; an entire facts of the Howe-Moore vanishing theorem for common semisimple Lie teams; a brand new remedy of Mautner's end result at the geodesic circulate of a Riemannian symmetric house; Mozes' end result approximately blending of all orders and the asymptotic distribution of lattice issues within the hyperbolic airplane; and Ledrappier's instance of a blending motion which isn't a blending of all orders.

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**Additional resources for Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces **

**Example text**

In order to study the generalized quadrangles which have some distinct axes of symmetry through some point p, we will introduce “Property (T)”. TGQ’s which satisfy Property (T) for some ordered ﬂag always have order (s, s2 ) for some s, and every TGQ of order (s, t) which has a (proper) subGQ of order s through the translation point satisﬁes Property (T) for some ordered ﬂag(s) containing the translation point. Property (T) is closely related to Property (G), and seems to be more general in the case of translation generalized quadrangles.

1 Chapter 3. The Known Generalized Quadrangles TGQ’s which Arise from Flocks (and their Translation Duals) Kantor semifield flock generalized quadrangles A ﬂock is called linear if all the ﬂock planes contain a common line. In that case, it can be shown that the ﬂock GQ is classical, and conversely. Let K be the quadratic cone with equation X0 X1 = X22 of PG(3, q), q odd. Then the q planes πt with equation tX0 − mtσ X1 + X3 = 0, where t ∈ GF(q), m is a given non-square in GF(q) and σ a given automorphism of GF(q), deﬁne a ﬂock F of K; see [120].

Thas [120]; ﬁrst given in explicit q-clan form in [81]) which is described in detail in [50]. However, it is rather involved and not as elegant as the BLT representation of T. Penttila in [98], which we will give here. Let K = GF(q) ⊆ GF(q 2 ) = K , q odd and q ≥ 5. Let η be a primitive element ¯, of K , and put η = ζ q−1 (so η has multiplicative order q + 1). Put T (x) = x + x where x ¯ = xq , and consider V = {(x, y, a) x, y ∈ K , a ∈ K} as a 5-dimensional vector space over K. Deﬁne a map Q : V → K by Q(x, y, a) = xq+1 + y q+1 − a2 .

### Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces by M. Bachir Bekka

by Thomas

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