Related papers: The Dehn Function of PSL(2,Z[1/p])

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL(2,K) representation variety, where K is the field of real or complex numbers, answering a…

In previous works, the universal mapping class group was taken to be the group PPSL(2,Z) of all piecewise PSL(2,Z) homeomorphisms of the unit circle S^1 with finitely many breakpoints among the rational points, and in fact, the Thompson…

In this note, we study the notion of random Dehn function and compute an asymptotic upper bound for finitely presented acylindrically hyperbolic groups whose Dehn function is at most polynomial. By showing that in these cases, if the group…

We discuss polyharmonic Maass forms of even integer weight on PSL(2, Z)\H, which are a generalization of classical Maass forms. We explain the role of real-analytic Eisenstein series E_k(z, s) and the differential operator…

We provide examples of finitely generated infinite covolume subgroups of $PSL(2,R)^r$ with a "big" limit set, e.g. that contains an open subset of the geometric boundary. They are given by the so called semi-arithmetic Fuchsian groups…

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar…

Let $P(x)$ be an irreducible quadratic polynomial in $\mathbb{Z}[x]$. We show that for almost all $n$, $P(n)$ does not lie in the range of Euler's totient function.

We prove a global version of the classical result that $p$-harmonic functions belong to $W^{2,2}_{loc}$ for $1<p<3+\frac{2}{n-2}$. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6].

This is essentially a translated (and explained) version of a peper Hecke published in 1930 where he shows, for a prime q, a relation between the class number h(-q) and the representation of PSL(2, Z / pZ) on the space of holomorphic…

The discreteness problem for finitely generated subgroups of $PSL(2,\mathbb{R})$ and $PSL(2,\mathbb{C})$ is a long-standing open problem. In this paper we consider whether or not this problem is decidable by an algorithm. Our main result is…

We discuss a special function (polyexponential) that extends the natural exponential function and also the exponential integral. The basic properties of the polyexponential are listed and some applications are given. In particular, it is…

Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by p. We remedy that defect in this paper by systematically applying…

We prove that the word problem of a finitely generated group $G$ is in NP (solvable in polynomial time by a non-deterministic Turing machine) if and only if this group is a subgroup of a finitely presented group $H$ with polynomial…

We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the…

Recently the author proved that the 1977 Hummel-Scheinberg-Zalcman conjecture on coefficients of nonvanishing $H^p$ functions is true for all $p = 2m, m \in \mathbb{N}$, i.e., for the Hilbertian Hardy spaces $H^{2m}$. As a consequence, this…

We describe all of the irreducible polynomial $\mathbb{F}_p\mathrm{SL}_2(p^r)$ representations which lift to $(\mathbb{Z}/p^s\mathbb{Z})\mathrm{SL}_2(p^r)$ representations for $s>1$, observing that they almost never do. We also show that…

Let $\overline{p}(n)$ denote the overpartition function. Liu and Zhang showed that $\overline{p}(a) \overline{p}(b)>\overline{p}(a+b)$ for all integers $a,b>1$ by using an analytic result of Engle. We offer in this paper a combinatorial…

An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the…

We prove an $L^p$ spectral multiplier theorem for functions of the $K$-invariant sublaplacian $L$ acting on the space of functions of fixed $K$-type on the group $SL(2,\mathbb{R}).$ As an application we compute the joint…

In a previous paper, I have defined non--commutative generalized Dedekind symbols for classical $PSL(2,Z)$--cusp forms using iterated period polynomials. Here I generalize this construction to forms of real weights using their iterated…