We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity of the first nonzero Steklov eigenvalues on trees: for finite trees G1 and G2, the first nonzero Steklov eigenvalue of G1 is greater than or equal to that of G2, provided that G1 is a subgraph of G2.

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For an infinite penny graph, we study the finitedimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharpdimensional estimate for the above spaces.M S C 2 0 2 0 05C10, 31C05 (primary)

For any proper action of a non-elementary group G on a proper geodesic metric space, we show that if G contains a contracting element, then there exists a sequence of proper quotient groups whose growth rate tends to the growth rate of G. Similar statements are obtained for a product of proper actions with contracting elements.The tools involved in this paper include the extension lemma for the construction of large tree, the theory of rotating families developed by F. Dahmani, V. Guirardel and D. Osin [10], and the contruction of a quasi-tree of metric spaces introduced by M. Bestvina, K. Bromberg and K. Fujiwara [5]. Several applications are given to CAT(0) groups and mapping class groups.