A locally compact group G is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure μ on G, either sup x∈G μ n (Cx) → 0 for all compact subsets C of G, or there exist x ∈ G and a compact subgroup K normalised by x such that μ n x −n → ω K , the normalised Haar measure on K. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors by compact groups invariant under the action and for factors by the connected component of the identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (pointwise distality). This gives some interesting corollaries to distality of certain actions and Choquet-Deny measures which actually motivated SCP and pointwise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup K such that G/K is distal and the conjugacy action of G on K is ergodic; moreover, if G itself is (pointwise) distal then G is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.Mathematics Subject Classification: Primary: 37B05, 22D05 Secondary: 22E15, 22D45

We study automorphisms α of a totally disconnected, locally compact group G which are expansive in the sense that n∈Z α n (U ) = {1} for some identity neighbourhood U ⊆ G. Notably, we prove that the automorphism induced by α on a quotient group G/N of G modulo an α-stable closed normal subgroup N is always expansive. Further results involve the contraction groups U α := {g ∈ G : α n (g) → 1 as n → ∞}. If α is expansive, then U α U α −1 is an open identity neighbourhood in G. We give examples where U α U α −1 fails to be a subgroup. However,Further results are devoted to the divisible and torsion parts of U α , and to the so-called "nub" U 0 = U α ∩U α −1 of an expansive automorphism.Classification: 22D05 (primary); 22D45, 22E20, 37A25, 37P20 (secondary)

We first study the growth properties of p-adic Lie groups and its connection with p-adic Lie groups of type R and prove that a non-type R p-adic Lie group has compact neighbourhoods of identity having exponential growth. This is applied to prove the growth dichotomy for a large class of p-adic Lie groups which includes p-adic algebraic groups. We next study p-adic Lie groups that admit recurrent random walks and prove the natural growth conjecture connecting growth and the existence of recurrent random walks, precisely we show that a p-adic Lie group admits a recurrent random walk if and only if it has polynomial growth of degree at most two. We prove this conjecture for some other classes of groups also. We also prove the Choquet-Deny Theorem for compactly generated p-adic Lie groups of polynomial growth and also show that polynomial growth is necessary and sufficient for the validity of the Choquet-Deny for all spread-out probabilities on Zariski-connected p-adic algebraic groups. Counter example is also given to show that certain assumptions made in the main results can not be relaxed. (2000): 20F15, 22E35, 60B15, 60J15, 60J45. Mathematics Subject Classification

Let G be a p-adic algebraic group of polynomial growth and H be a closed subgroup of G. We prove the growth conjecture for the homogeneous space G/H, that is, G/H supports a recurrent random walk if and only if G/H has polynomial growth of degree atmost two. Mathematics Subject Classification (2000). Primary 60G50 ; Secondary 60B15, 22D05.