Many mayors quietly worry that their cities will turn into nightmarish Detroits, with a rotten core and a choice collection of the region's most troubled residents. Mayors are forming alliances with nearby settlements mostly because they have to. Few cities can now expect to dominate their hinterlands simply by virtue of being big. Across America suburbs are strongly competing for people, offices and cultural centres.

Until recently Louisville seemed to be following the path of many industrial cities. Its factories were shedding workers. Middle-class whites were drifting to the suburbs and beyond. Between 1960 and 2000 the city's population dropped from 391,000 to 256,000. For the city to prosper, Mr Abramson realised, it must work with its neighbours. Ever since he took office the relationship has become closer.

In 2003 Louisville joined forces with surrounding Jefferson county in the biggest such merger since the 1970s (Indianapolis and Nashville, for example, also have consolidated city-county governments). Mr Abramson, who had served his three terms as city mayor, easily won the top job in the new “Louisville Metro”. Since then he has streamlined public services and accelerated the redevelopment of downtown Louisville.

Greater Louisville Inc, the metropolitan chamber of commerce, has helped companies like Geek Squad, a computer-service outfit, move beyond the city's borders. “It's better that a company locate in the next county over,” Mr Abramson says, “than we lose it to Chicago or Atlanta, Nashville or Cincinnati.”

Increasingly, co-operation extends across the Ohio river. At present Louisville and Indiana are linked by two ageing, congested bridges—a big problem for a city that claims to be a logistics hub. Greater Louisville Inc has joined with its equivalent in southern Indiana to promote plans for two new bridges, and is considering whether, and how, to share revenue from a new industrial park in southern Indiana.

Tucson is one of the sunbelt's rustiest cities. Its population is growing much more slowly than that of Phoenix, to the north, or Albuquerque to the east. It is poorer and more Hispanic than its surroundings. Tucson accounts for just over half the population of 9,000-square-mile Pima county. But probably not for long: between 2000 and 2006 some 71% of the county's population growth and more than half of its job growth took place outside the city. “We get all the negative externalities and none of the positive ones, except for sales taxes,” complains Mike Hein, the city manager.

Bob Walkup, Tucson's mayor, has helped end a long struggle between the city and county governments. They now work closely to attract new businesses and write transport policy. Like Mr Abramson, Mr Walkup is now focused on downtown.

Tucson's core is singularly bleak and unimpressive. But the city has extracted tax concessions from the state that will help it to build a hotel and an arena, and expand its convention centre. City mayors have proved they can work with suburbanites to solve common problems. Their next challenge is to convince them that both will be stronger with a beating heart.