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The rate of convergence of simple random walk on the Heisenberg group over $Z/nZ$ with a standard generating set was determined by Bump et al [1,2]. We extend this result to random walks on the same groups with an arbitrary minimal symmetric generating set. We also determine the rate of convergence of simple random walk on higher-dimension
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Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with Peres conjecture in [3] relating sharp cut off, spectral gap, and mixing time.
We analyse the possible ways of gluing twisted products of circles with asymptotically cylindrical Calabi-Yau manifolds to produce manifolds with holonomy G_2, thus generalising the twisted connected sum construction of Kovalev and Corti, Haskins, Nordstrom, Pacini. We then express the extended nu-invariant of Crowley, Goette, and Nordstrom in terms of fixpoint and gluing contributions, which include different types of (generalised) Dedekind sums. Surprisingly, the calculations involve some non-trivial number-theoretical arguments connected with special values of the Dedekind eta-function and the theory of complex multiplication. One consequence of our computations is that there exist compact G_2-manifolds that are not G_2-nullbordant.
We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different suitably unrelated and generic polynomials. The proofs combine analytic arguments with the algebraic interpretation of exponential sums and their monodromy groups.
Let $mathcal{T}$ be a rooted tree endowed with the natural partial order $preceq$. Let $(Z(v))_{vin mathcal{T}}$ be a sequence of independent standard Gaussian random variables and let $alpha = (alpha_k)_{k=1}^infty$ be a sequence of real numbers with $sum_{k=1}^infty alpha_k^2<infty$. Set $alpha_0 =0$ and define a Gaussian process on $mathcal{T}$ in the following way: [ G(mathcal{T}, alpha; v): = sum_{upreceq v} alpha_{|u|} Z(u), quad v in mathcal{T}, ] where $|u|$ denotes the graph distance between the vertex $u$ and the root vertex. Under mild assumptions on $mathcal{T}$, we obtain a necessary and sufficient condition for the almost sure boundedness of the above Gaussian process. Our condition is also necessary and sufficient for the almost sure uniform convergence of the Gaussian process $G(mathcal{T}, alpha; v)$ along all rooted geodesic rays in $mathcal{T}$.
Let $M_n$ be the connect sum of $n$ copies of $S^2 times S^1$. A classical theorem of Laudenbach says that the mapping class group $text{Mod}(M_n)$ is an extension of $text{Out}(F_n)$ by a group $(mathbb{Z}/2)^n$ generated by sphere twists. We prove that this extension splits, so $text{Mod}(M_n)$ is the semidirect product of $text{Out}(F_n)$ by $(mathbb{Z}/2)^n$, which $text{Out}(F_n)$ acts on via the dual of the natural surjection $text{Out}(F_n) rightarrow text{GL}_n(mathbb{Z}/2)$. Our splitting takes $text{Out}(F_n)$ to the subgroup of $text{Mod}(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbachs original proof, including the identification of the twist subgroup with $(mathbb{Z}/2)^n$.
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