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In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative; consequently, these examples are not linear. Let S be a connected orientable surface without boundary that is of infinite type, so that π 1 (S) is infinitely generated. The mapping class group Map(S) is the group of homotopy classes of orientation-preserving homeomorphisms of S. The mapping class group of an infinite-type surface is often called a big mapping class group. Similarly, let S g be the compact connected orientable surface of genus g and let Map(S g) be its mapping class group. In this note we address several questions about subgroups of Map(S). We first prove some results about the triviality of centers of subgroups of Map(S). We then show that Map(S) never satisfies the strong Tits alternative enjoyed by Map(S g), as well as some related results. Centers. The center of Map(S g) is trivial for g ≥ 3, while its center is a cyclic group generated by the hyperelliptic involution when g = 1 or 2 ([4, Section 3.4]). The centers of mapping class groups of finite-type surfaces with punctures and boundary components were computed by Paris-Rolfsen ([12, Theorem 5.6]). Although Dehn twists about boundary components are always

Abstract. We compute the monoid V[L K (E)] of isomorphism classes of finitely generated projective modules of a Leavitt path algebra over an arbitrary directed graph. Our result generalizes the result of Ara, Moreno, and Pardo in which they computed the monoid V[L K (E)] of a Leavitt path algebra over a countable row-finite directed graph.

Classical parking functions are defined as the parking preferences for n cars driving (from west to east) down a one-way street containing parking spaces labeled from 1 to n (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the n-tuple containing the cars' parking preferences a parking function.In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to k spaces west of their preferred spot to park before proceeding east if all of those k spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule k-Naples parking functions of length n. This generalization gives a natural interpolation between classical parking functions, the case when k = 0, and all n-tuples of positive integers 1 to n, the case when k ≥ n − 1. Our main result provides a recursive formula for counting k-Naples parking functions of length n. We also give a characterization for the k = 1 case by introducing a new function that maps 1-Naples parking functions to classical parking functions, i.e. 0-Naples parking functions. Lastly, we present a bijection between k-Naples parking functions of length n whose entries are in weakly decreasing order and a family of signature Dyck paths.

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