Centers of subgroups of big mapping class groups and the Tits alternative

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 and give a proof that every countable group appears as a subgroup of some big mapping class group.Comment: 6 pages, 1 figur

Graphs of curves for surfaces with finite-invariance index (1)

In this note we make progress toward a conjecture of Durham–Fanoni–Vlamis, showing that every infinite-type surface with fi­ni­te-invariance index (1) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by Mann–Rafi in their study of the coarse geometry of big mapping class groups

Centers of subgroups of big mapping class groups and the Tits alternative

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

Adding a point to configurations in closed balls

We answer the question of when a new point can be added in a continuous way to configurations of n distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of n points if and only if n ≠ 1. On the other hand, when the points are not ordered and the dimension of the ball is at least 2, a point can be added continuously if and only if n = 2. These results generalize the Brouwer fixed-point theorem, which gives the negative answer when n = 1. We also show that when n = 2, there is a unique solution to both the ordered and unordered versions of the problem up to homotopy

Determining in-channel (dead zone) transient storage by comparing solute transport in a bedrock channel–alluvial channel sequence, Oregon

Current stream tracer techniques do not allow separation of in-channel dead zone (e.g., eddies) and out-of-channel (hyporheic) transient storage, yet this separation is important to understanding stream biogeochemical processes. We characterize in-channel transient storage with a rhodamine WT solute tracer experiment in a 304 m cascade-pool-type bedrock reach with no hyporheic zone. We compare the solute breakthrough curve (BTC) from this reach to that of an adjacent 367 m alluvial reach with significant hyporheic exchange. In the bedrock reach, transient storage has an exponential residence time distribution with a mean residence time of 3.0 hours and a ratio of transient storage to stream volume of 0.14, demonstrating that at moderate discharge, bedrock in-channel storage zones provide a small volume of transient storage with substantial residence time. In the alluvial reach, though pools are similar in size to those in the bedrock reach, transient storage has a power law residence time distribution with a mean residence time of >100 hours (estimated at nearly 1200 hours) and a ratio of storage to stream volume of 105. Because the in-channel hydraulics of bedrock reaches are simpler than alluvial step-pool reaches, the bedrock results are probably a lower end-member with respect to volume and residence time, though they demonstrate that in-channel storage may be appreciable in some reaches. These results suggest that in-stream dead zone transient storage may be accurately simulated by exponential RTDs but that hyporheic exchange is better simulated with a power law RTD as a consequence of more complicated flow path and exchange dynamics.Keywords: residence time distribution, transient storage, dead zone, hyporheic exchange, H.J. Andrews Experimental Forest, bedrock channelKeywords: residence time distribution, transient storage, dead zone, hyporheic exchange, H.J. Andrews Experimental Forest, bedrock channe