Given a simple, unweighted, undirected graph G=(V,E) with |V|=n and |E|=m, and parameters 0 < \varepsilon, \delta <1, along with Degree, Neighbour, Pair and RandomEdge query access to G, we provide a query-based randomized algorithm to generate an estimate \widehat{T} of the number of triangles T in G, such that \widehat{T} \in [(1-\varepsilon)T , (1+\varepsilon)T] with probability at least 1-\delta. The query complexity of our algorithm is \widetilde{O}\left({m \alpha \log(1/\delta)}/{\varepsilon^3 T}\right), where \alpha is the arboricity of G. Our work can be seen as a natural progression to the line of recent works [Eden et al., SIAM J Comp., 2017; Assadi et al., ITCS 2019; Eden et al., SODA 2020] that considered subgraph or triangle counting with or without the use of RandomEdge query. Of these works, Eden et al. [SODA 2020] considers the role of arboricity. Our work is the first to consider how RandomEdge query can leverage the structural property of arboricity. Furthermore, continuing in the line of work of Assadi et al. [APPROX/RANDOM 2022], we also provide a lower bound of \widetilde{\Omega}\left({m \alpha \log(1/\delta)}/{\varepsilon^2 T}\right) that matches the upper bound exactly on arboricity, \delta, and almost on \varepsilon.

We shall discuss counting reversible elements in the Picard group. This group exhibits a higher count than equidistribution would allow. This exceptional feature can be explained partly by its action on hyperbolic 3-space and partly by its arithmetic. We'll try to explore this and also compare it with the modular group which acts on the hyperbolic 2-space. The counting includes joint work with Debattam Das and Krishnendu Gangopadhyay.