### A note on a multi-variable polynomial link invariant

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We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in [CGHM] and [A].

We define for each group G the skein algebra of G. We show how it is related to the Kauffman bracket skein modules. We prove that skein algebras of abelian groups are isomorphic to symmetric subalgebras of corresponding group rings. Moreover, we show that, for any abelian group G, homomorphisms from the skein algebra of G to C correspond exactly to traces of SL(2,C)-representations of G. We also solve, for abelian groups, the conjecture of Bullock on SL(2,C) character varieties of groups - we show...

We show that the ${L}^{p}$ boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

Let Ω be an open subset of ${\mathbb{R}}^{d}$ with 0 ∈ Ω. Furthermore, let ${H}_{\Omega}=-{\sum}_{i,j=1}^{d}{\partial}_{i}{c}_{ij}{\partial}_{j}$ be a second-order partial differential operator with domain ${C}_{c}^{\infty}\left(\Omega \right)$ where the coefficients ${c}_{ij}\in {W}_{loc}^{1,\infty}\left(\Omega \u0305\right)$ are real, ${c}_{ij}={c}_{ji}$ and the coefficient matrix $C=\left({c}_{ij}\right)$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ${\int}_{0}^{\infty}ds{s}^{d/2}{e}^{-\lambda \mu \left(s\right)\xb2}<\infty $ for some λ > 0 where $\mu \left(s\right)={\int}_{0}^{s}dtc{\left(t\right)}^{-1/2}$ then we establish that ${H}_{\Omega}$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover...

We consider pure mth order subcoercive operators with complex coefficients acting on a connected nilpotent Lie group. We derive Gaussian bounds with the correct small time singularity and the optimal large time asymptotic behaviour on the heat kernel and all its derivatives, both right and left. Further we prove that the Riesz transforms of all orders are bounded on the Lp -spaces with p ∈ (1, ∞). Finally, for second-order operators with real coefficients we derive matching Gaussian lower bounds...

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